Operational amplifiers

The operational amplifier, or op amp, is the most widely used analog building block in electronics. It is an integrated circuit containing dozens of transistors, but engineers rarely need to think about them. From the outside, an op amp is a very high-gain amplifier of the difference between two input voltages, and with a few resistors around it, it becomes an amplifier of almost any precise gain, a buffer, a summer, a subtractor, a filter, or a converter from current to voltage. The name comes from its original use in analog computers in the 1940s, where op amps performed mathematical operations such as addition and integration.

Op amps appear in every sensor interface, audio system, power supply controller, battery management system and measurement instrument. Inside a data center they condition the signals from temperature, current and pressure sensors, and in optical receivers a close relative, the transimpedance amplifier, converts the tiny photodiode current from a fiber into a voltage. This chapter introduces the ideal op amp model, works through the standard circuits with the methods of Lessons 2 and 4, and explains the practical limits of real devices.

The op amp and its model

An op amp has two inputs, the non-inverting input (marked +) and the inverting input (marked −), and one output. It also has two power supply connections, often \(+V_{CC}\) and \(-V_{EE}\) (for example ±15 V), or a single supply and ground, which are usually left off circuit diagrams but are always there. The output voltage is measured relative to ground.

An op amp amplifies the difference between its inputs:

where \(A\) is the open-loop gain, typically \(10^5\) to \(10^6\) or more. The output cannot exceed the supply voltages: when the formula asks for more, the output saturates at a level near the supply rail (within a volt or two for older designs, within millivolts for "rail-to-rail" designs). The resulting transfer characteristic has a very steep linear region and two flat saturation regions.

The ideal op amp

With negative feedback, a path from the output back to the inverting input, the enormous gain works for us. The output settles at whatever voltage makes \(v_+ - v_-\) almost exactly zero, because any difference is amplified \(A\) times and corrected. This leads to the ideal op amp model, which predicts real op-amp circuits accurately in most applications:

• Rule 1, no input current: the inputs draw no current, \(i_+ = i_- = 0\) (real input currents are nanoamperes or picoamperes).
• Rule 2, virtual short: with negative feedback, the two input voltages are equal, \(v_+ = v_-\). The inputs are not actually connected (no current flows between them), which is why it is called a "virtual" short.
• The output can supply whatever current the circuit needs (within limits), so its output resistance is effectively zero.

Analysis of an ideal op amp circuit uses only these rules plus KCL at the input nodes. One warning: never write KCL at the op amp's output node. The output current comes from the power supplies through pins not shown on the diagram, so it is unknown and KCL there gives no useful information.

The inverting amplifier

In the inverting amplifier, the input signal drives the inverting input through a resistor \(R_1\), a feedback resistor \(R_f\) connects the output back to the inverting input, and the non-inverting input is grounded.

By the virtual short, \(v_- = v_+ = 0\): the inverting input is a virtual ground. The current through \(R_1\) is \(v_{in}/R_1\). Since no current enters the op amp, all of it flows through \(R_f\), producing a drop of \(R_f v_{in}/R_1\) from the inverting input (0 V) to the output. So:

The gain depends only on the ratio of two resistors, not on the op amp's own gain, which varies from part to part and with temperature. That is the magic of negative feedback: precise, stable gain from an imprecise amplifier. The minus sign means the output is inverted. The input resistance seen by the source is simply \(R_1\), because the inverting input sits at 0 V.

The non-inverting amplifier and the voltage follower

In the non-inverting amplifier, the signal drives the non-inverting input directly, and the feedback network \(R_f\) and \(R_1\) forms a divider from the output to ground, with its midpoint at the inverting input.

The divider gives \(v_- = v_o R_1/(R_1 + R_f)\), and the virtual short sets \(v_- = v_{in}\). Solving:

The gain is positive and at least 1. Because the input drives the op amp input directly and draws no current, the input resistance is essentially infinite, so the amplifier does not load the source at all.

Setting \(R_f = 0\) and removing \(R_1\) gives the voltage follower (buffer): the output is connected straight to the inverting input, and \(v_o = v_{in}\). A gain of 1 sounds useless, but the follower draws no current from its source and can drive a heavy load from its low-resistance output. It is the cure for the loading problem of Lesson 2.

Summing and difference amplifiers

The summing amplifier

Connecting several inputs, each through its own resistor, to the virtual ground of an inverting amplifier adds their currents. KCL at the inverting input (at 0 V) gives:

Each input is weighted by its own resistor, and because the summing point is a virtual ground, the inputs do not interact. Audio mixing consoles combine channels this way. With binary-weighted resistors, a summing amplifier becomes a simple digital-to-analog converter.

The difference amplifier

Many sensors produce a small signal riding on a large common voltage, for example the two midpoints of a Wheatstone bridge, each at about half the supply voltage. The difference amplifier responds only to the difference. With \(v_1\) driving the inverting input through \(R_1\) (feedback \(R_2\)), and \(v_2\) driving the non-inverting input through a matching divider of \(R_1\) and \(R_2\) to ground:

The voltage common to both inputs is rejected. The quality of that rejection, the common-mode rejection ratio (CMRR), depends on how well the resistor ratios match: a 0.1% mismatch can let a little of the common voltage through. Instrumentation amplifiers, which add input buffers to a difference amplifier, give very high input resistance and excellent rejection and are the standard front end for bridge sensors, strain gauges and thermocouples.

Current to voltage: the transimpedance amplifier

Some sensors produce a current rather than a voltage. A photodiode, for example, produces a current proportional to the light falling on it. Feeding that current into the virtual ground of an op amp with only a feedback resistor gives \(v_o = -i_{in}R_f\). The gain has units of ohms (volts per ampere), hence the name transimpedance amplifier (TIA). A photodiode current of 2 µA into a 1 MΩ feedback resistor gives −2 V. Every optical transceiver in a data center network has a high-speed transimpedance amplifier immediately behind its photodiode, converting microamperes of photocurrent into a voltage that the receiver can decode.

Analyzing op amp circuits systematically

More complex op amp circuits are analyzed with nodal analysis plus the two ideal rules. Write KCL at the op amp input nodes (using \(i_+ = i_- = 0\)) and at any other nodes, use \(v_+ = v_-\) as an extra equation, and never write KCL at an op amp output. Cascaded stages can be analyzed one at a time, because each ideal op amp output is a voltage source unaffected by what it drives: the overall gain is the product of the stage gains.

Real op amps

The ideal model is accurate when the circuit gain is far below the open-loop gain and the signals are slow. Real op amps depart from it in several ways; Lesson 9 quantifies the DC errors, and later courses explore the rest:

• Finite open-loop gain. For a non-inverting amplifier with feedback fraction \(\beta = R_1/(R_1 + R_f)\), the exact gain is \(A/(1 + A\beta)\). With \(A = 200{,}000\) and an ideal gain of 4 (\(\beta = 0.25\)), the exact gain is 3.99992, an error of 0.002%. The ideal rules are excellent here.
• Limited bandwidth. Open-loop gain falls with frequency. Many op amps have a roughly constant gain-bandwidth product: a 1 MHz op amp used at a gain of 4 has a bandwidth of about 250 kHz.
• Slew rate. The output can change only so fast (for example 0.5 V/µs for older general-purpose parts), which distorts large, fast signals.
• Offsets and bias currents. Small built-in input voltage offsets (microvolts to millivolts) and input currents (picoamperes to nanoamperes) produce output errors, important in precision measurement.
• Output limits. The output saturates near the supply rails and can only supply limited current, typically tens of milliamperes.

Datasheets specify all of these, and choosing an op amp means matching them to the application: precision and low offset for sensors, high bandwidth and slew rate for video and communications, low power for battery devices.