# 1.1: Introduction

Before we can begin our study of the operational amplifier, it is very important that certain background elements be in place. The purpose of this chapter is to present the very useful analysis concepts and tools associated with the decibel measurement scheme and the frequency domain. We will also be examining the differential amplifier that serves as the heart of most operational amplifiers. With a thorough working knowledge of these items, you will find that circuit design and analysis will proceed at a much quicker and more efficient pace. Consider this chapter as an investment in time, and treat it appropriately.

The decibel measurement scheme is in very wide use, particularly in the field of communications. We will be examining its advantages over the ordinary system of measurement, and how to convert values of one form into the other. One of the more important parameters of a circuit is its frequency response. To this end, we will be looking at the general frequency domain representations of a circuit’s gain and phase. This will include both manual and computer generated analysis and graphing techniques. While our main emphasis will eventually concentrate on application with operational amplifiers, the techniques explored can be applied equally to discrete circuits. Indeed, our initial examples will use simple discrete or black box circuits exclusively.

Finally, we will examine the DC and AC operation of the differential amplifier. This is an amplifier that utilizes two active devices and offers dual inputs. It offers certain features that make it suitable as the first section of most operational amplifiers.

## 1.1.1: Variable Naming Convention

One item that often confuses students of almost any subject is nomenclature. It is important, then, that we decide upon a consistent naming convention at the outset. Throughout this text, we will be examining numerous circuits containing several passive and active components. We will be interested in a variety of parameters and signals. Although we will utilize the standard conventions, such as $$f_c$$ for critical frequency and $$X_c$$ for capacitive reactance, a great number of other possibilities exist. In order to keep confusion to a minimum, we will use the following conventions in our equations for naming devices and signals that haven’t been standardized.

 $$R$$ Resistor (DC, or actual circuit component) $$r$$ Resistor (AC equivalent, where phase is 0 or ignored) $$C$$ Capacitor $$L$$ Inductor $$Q$$ Transistor (Bipolar or FET) $$D$$ Diode $$V$$ Voltage (DC) $$v$$ Voltage (AC) $$I$$ Current (DC) $$i$$ Current (AC)

Resistors, capacitors and inductors are differentiated via a subscript that usually refers to the active device it is connected to. For example, $$R_E$$ is a DC bias resistor connected to the emitter of a transistor, while $$r_C$$ refers to the AC equivalent resistance seen at a transistor’s collector. $$C_E$$ refers to a capacitor connected to a transistor’s emitter lead (most likely a bypass or coupling capacitor). Note that the device related subscripts are always shown in upper case, with one exception: If the resistance or capacitance is part of the device model, the subscript will be shown in lower case to distinguish it from the external circuit components. For example, the AC dynamic resistance of a diode would be called $$r_d$$. If no active devices are present, or if several items exist in the circuit, a simple numbering scheme is used, such as $$R_1$$. In very complex circuits, a specific name will be given to particularly important components, as in $$R_{source}$$.

Voltages are normally given a two-letter subscript indicating the nodes at which it is measured. $$V_{CE}$$ is the DC potential from the collector to the emitter of a transistor, while $$v_{BE}$$ indicates the AC signal appearing across a transistor’s base-emitter junction. A single-letter subscript, as in $$V_B$$, indicates a potential relative to ground (in this case, base to ground potential). The exceptions to this rule are power supplies, that are given a double letter subscript indicating the connection point ($$V_{CC}$$ is the collector power supply), and particularly important potentials that are directly named, as in $$v_{in}$$ (AC input voltage) and $$V_{R1}$$ (DC voltage appearing across $$R_1$$). If an Equation for a specific potential is valid for both the AC and DC equivalent circuits, the uppercase form is preferred (this makes things much more consistent with the vast majority of op amp circuits that are directly coupled, and thus can amplify both AC and DC signals). Currents are named in a similar way, but generally use a single subscript referring to the measurement node ($$I_C$$ is the DC collector current). All other items are directly named. By using this scheme, you will always be able to determine whether the item expressed in an Equation is a DC or AC equivalent, its approximate circuit location, and other factors about it.