Chapter 1
Introduction

We start with the iconic figure (Figure 1.1), which depicts a bulb connected to a battery. Whenever the loop is closed and a full connection is established, the bulb comes on and starts to consume energy provided by the battery. The process is often described as the conversion of the chemical energy stored in the battery into electrical energy that is further released as heat and light by the bulb. The connection between the bulb and battery consists of two wires between the positive and negative terminals of the bulb and battery. These wires are shown as simple straight lines, whereas in real life they are usually coaxial or paired cables that are isolated from the environment.

Figure 1.1 A simple circuit involving a bulb connected to a battery. The connection between the bulb and battery is shown via simple lines.

The purpose of this first chapter is to introduce basic concepts of electrical circuits. In order to understand circuits, such as the one above, we first need to understand electric charge, potential, and current. These concepts provide a basis for recognizing the interactions between electrical components. We further discuss electric energy and power as fundamental variables in circuit analysis. The time and frequency in circuits, as well as related limitations, are briefly considered. Finally, we study conductivity and resistance, as well as resistors, independent sources, and dependent sources as common components of basic circuits.

1.1 Circuits and Important Quantities

An electrical circuit is a collection of components connected via metal wires. Electrical components include but are not limited to resistors, inductors, capacitors, generators (sources), transformers, diodes, and transistors. In circuit analysis, wire shapes and geometric arrangements are not important and they can be changed, provided that the connections between the components remain the same with fixed geometric topology. Wires often meet at intersection points; a connection of two or more wires at a point is called a node. Before discussing how circuits can be represented and analyzed, we first need to focus on important quantities, namely, electric charge, electric potential, and current, as well as energy and power.

Figure 1.2 A circuit involving connections of four components labeled from to . From the circuit-analysis perspective, connection shapes are not important, and these three representations are equivalent.

1.1.1 Electric Charge

Electric charge is a fundamental property of matter to describe force interactions among particles. According to Coulomb's law, there is an attractive (negative) force between a proton and an electron given by

which is significantly larger than (around times) the gravity between these particles. In the above, is the distance between the proton and electron, given in meters (m). This law can be rewritten by using Coulomb's constant

as

where

are the electrical charges of the proton and electron, respectively, in units of coulombs (C). Coulomb's constant enables the generalization of the electric force between any arbitrary charges and as

where and are assumed to be point charges (theoretically squeezed into zero volumes), which are naturally formed of collections of protons and electrons.

The definition of the electric force above requires at least two charges. On the other hand, it is common to extend the physical interpretation to a single charge. Specifically, a stationary charge is assumed to create an electric field (intensity) that can be represented as

where is now the distance measured from the location of the charge. This electric field is in the radial direction, either outward (positive) or inward (negative), depending on the type (sign) of the charge. Therefore, we assume that an electric field is always formed whether there is a second test charge or not. If there is at a distance , the electric force is now measured as

either as repulsive (if and have the same sign) or attractive (if and have different signs).

Figure 1.3 Electric field lines created individually by a positive charge and a negative charge. An electric field is assumed to be created whether there is a second test charge or not. If a test charge is located in the field, repulsive or attractive force is applied on it.

The definition of the electric field is so useful that, in many cases, even the sources of the field are discarded. Consider a test charge exposed to some electric field . The force on can be calculated as

without even knowing the sources creating the field. This flexibility further allows us to define the electric potential concept, as discussed below.

1.1.2 Electric Potential (Voltage)

Consider a charge in some electric field created by external sources. Moving the charge from a position to another position may require energy if the movement is opposite to the force due to the electric field. This energy can be considered to be absorbed by the charge. If the movement and force are aligned, however, energy is extracted from the charge. In general, the path from to may involve absorption and release of energy, depending on the alignment of the movement and electric force from position to position. In any case, the net energy absorbed/released depends on the start and end points, since the electric field is conservative and its line integral is path-independent.

Figure 1.4 Movement of a charge in an electric field created by external sources. The energy absorbed or released by the charge does not depend on the path but depends on the potentials at the start and end points. The electric potential (voltage) is always defined between two points, while selecting a reference point as a ground enables unique voltage definitions at all points.

Electric potential (voltage) is nothing but the energy considered for a unit charge (1 C) such that it is defined independent of the testing scheme. Specifically, the work done in moving a unit charge from a point to another point is called the voltage between and . Conventionally, we have

as the voltage between and , corresponding to the work done in moving the charge from to . If , then work must be done to move the charge (the energy of the charge increases). On the other hand, if , then the work done is negative, indicating that energy is actually released due to the movement of the charge. The unit of voltage is the volt (V), and 1 volt is 1 joule per coulomb (J/C).

A proper voltage definition always needs two locations and a polarity definition. Considering three separate points , , and , we have

and

The equality above is a result of the conservation of the electric energy (conservative electric field). On the other hand, , , and are not yet uniquely defined. In order to simplify the analysis in many cases, a location can be selected as a reference with zero potential. In circuit analysis, such a location that corresponds to a node is called ground, and it allows us to define voltages at all other points uniquely. For example, if in the above, we have .

1.1.3 Electric Current

A continuous movement of electric charges is called electric current. Conventionally, the direction of a current flow is selected as the direction of movement of positive charges. The unit of current is the ampere (A), and 1 ampere is 1 coulomb per second (C/s). Formally, we have

where and represent charge and time, respectively. The current itself may depend on time, as indicated in this equation. But, in some cases, we only have steady currents, , where does not depend on time.

Different types of current exist, as discussed in Section 1.2.1. In circuit analysis, however, we are restricted to conduction currents, where free electrons of metals (e.g., wires) are responsible for current flows. Since electrons have negative charges and an electric current is conventionally defined as the flow of positive charges, electron movements and the current direction on a wire are opposite to each other. Indeed, when dealing with electrical circuits, using positive current directions is so common that the actual movement of charges (electrons) is often omitted.

When charges move, they interact with each other differently such that they cannot be modeled only with an electric field. For example, two parallel wires carrying currents in opposite directions attract each other, even though they do not possess any net charges considering both electrons and protons. Similar to the interpretation that electric field leads to electric force, this attraction can be modeled as a magnetic field created by a current, which acts as a magnetic force on a test wire. Electric and magnetic fields, as well as their coupling as electromagnetic waves, are described completely by Maxwell's equations and are studied extensively in electromagnetics.

Figure 1.5 On a metal wire, the conventional current direction, which is defined as positive charge flow, is the opposite of the actual electron movements. In a circuit, voltages are defined at the nodes, as well as across components, using the sign convention.

1.1.4 Electric Voltage and Current in Electrical Circuits

In electrical circuit analysis, charges, fields, and forces are often neglected, while electric voltage and electric current are used to describe all phenomena. This is completely safe in the majority of...