Chapter 1 introduced basic concepts such as current, voltage, and power in an electric circuit. To actually determine the values of this variable in a given circuit requires that we understand some fundamental laws govern electric circuits. These laws known as Ohm’s law and Kirchhoff’s laws, from the foundation upon which electric circuit analysis is build.

In this chapter, in addition to these laws we shall discuss some techniques commonly applied in circuit design and analysis. These techniques include combining resistors in series or parallel, voltage division, current division and delta-to-wye and wye-to-delta transformations

Materials in general have a characteristic behavior of resisting the flow of electric charge. This physical property or ability to resist current known as resistance and is represented by the symbol R. the resistance of any material with a uniform cross-sectional area A depends on A and its length l, as shown in Figure 1(a). We can represent resistance (as measured in the laboratory), in mathematical form,

where ρρ size 12{ρ} {} is known as the resistivity of the material in ohm-meters. Good conductors, such as copper and aluminum, have low resistivities, while insulators, such as mica and paper, have high resistivities. Table 1 presents the values of ρρ size 12{ρ} {} for some common materials and shows which materials are used for conductors, insulators, and semiconductors.

The circuit element used to model the current-resisting behavior of a material is the resistor. For the purpose of constructing circuits, resistors are usually made from metallic alloys and carbon compounds. The circuit symbol for the resistor is shown in Figure 1(b), where R stands for the resistance of the resistor. The resistor is the simplest passive element.

Georg Simon Ohm (1787-1854), a German physicist, is credited with finding the relationship between current and voltage for a resistor. This relationship is known as Ohm’s law.

That is,

Ohm defined the constant of proportionality for a resistor to be the resistance, R. Thus, Equation 2 becomes

Which is the mathematical form of Ohm’s law. R in Equation 3 is measured in the unit of Ohms, designated ΩΩ size 12{ %OMEGA } {}. Thus,

We may deduce from Equation 3 that

So that

1 ΩΩ size 12{ %OMEGA } {} = 1 V/A

To apply Ohm’s law as stated in Equation 3, we must pay careful attention to the current direction and voltage polarity. The direction of current i and the polarity of voltage v must conform with the passive sign convention, as shown in Figure 1(b). This implies that current flows from a higher potential to a lower potential in order for v = iR. If current flows from a lower potential to a higher potential, v = -iR.

Since the value of R can range from zero to infinity, it is important that we consider the two extreme possible values of R. An element with R = 0 is called a short circuit, as shown in Figure 2(a). For a short circuit,

Showing that the voltage is zero but the current could be anything. In practice, a short circuit is usually a connecting wire assumed to be a perfect conductor. Thus,

Similarly, an element with R = ∞∞ size 12{ infinity } {} is known as an open circuit, as shown in Figure 2(b). For an open circuit,

Indicating that the current is zero though the voltage could be anything. Thus,

A resistor is either fixed or variable. Most resistors are of the fixed type, meaning their resistance remains constant. The two common types of fixed resistors (wirewound and composition) are shown in Figure 3. The composition resistors are used when large resistance is needed. The circuit symbol in Figure 3(b) is for a fixed resistor. Variable resistors have adjustable resistance. The symbol for a variable resistor is shown in Figure 4(a). A common variable resistor is known as a potentiometer or pot for short, with the symbol shown in Figure 4(b). The pot is three-terminal element with the sliding contact or wiper. By the sliding wiper, the resistances between the wiper terminal and the fixed terminals vary. Like fixed resistors, variable resistors can either be of wirewound or composition type, as shown in Figure 5. Although resistors like those in Figure 3 and Figure 5 are used in circuit designs, today most circuit components including resistors are either surface mounted or integrated, as typically shown in Figure 6.

It should be pointed out that not all resistors obey Ohm’s law. A resistor that obeys Ohm’s law is known as a linear resistor. It has a constant resistance and thus its current-voltage characteristic is as illustrated in Figure 7(a): its i-v graph is a strait passing through the origin. A nonlinear resistor does not obey Ohm’s law. Its resistance varies with current and its i-v characteristic is typically shown in Figure 7(b). Examples of devices with nonlinear resistance are the lightbulb and the diode. Although all practical resistors may exhibit nonlinear behavior under certain conditions, we will assume in this book that all elements actually designated as resistors are linear.

A useful quantity in circuit analysis is the reciprocal of resistance R, known as conductance and denoted by G:

The conductance is a measure of how well an element will conduct electric current. The unit of conductance is the siemens (S), the SI unit of conductance:

Thus,

The same resistance can be expressed in ohms or siemens. For example, 10 ΩΩ size 12{ %OMEGA } {} is the same as 0.1 S. from Equation 7, we may write

The power dissipated by a resistor can be expressed in term of R. using (Reference) and Equation 3,

The power dissipated by a resistor may also be expressed in terms of G as

We should note two things from Equation 10 and Equation 11:

1. The power dissipated in a resistor is a nonlinear function of either current or voltage.

2. Since R and G are positive quantities, the power dissipated in a resistor is always positive. Thus, a resistor always absorbs power from the circuit. This confirms the idea that a resistor is a passive element, incapable of generating energy.

Since the elements of an electric circuit can be interconnected in several ways, we used to understand some basic concepts of network topology. To differentiate between a circuit and a network, we may regard a network as an interconnection of elements or devices, whereas a circuit is a network providing one or more closed paths. The convention, when addressing network topology, is to use the word network and circuit mean the same thing when used in this context. Such elements include branches, nodes, and loops.

In the other words, a branch represents any two-terminal element. The circuit in Figure 8 has five branches, namely, the 10 V voltage source, the 2 A current source, and the three resistors.

A node is usually indicated by a dot in a circuit. If a short circuit (a connecting wire) connects two nodes, the two nodes constitute a single node. The circuit in Figure 8 has three nodes a, b, and c. notice that the three points that form node b are connected by perfectly conducting wires and therefore constitute a single point. The same is true of the four points forming nodes c. we demonstrate that the circuit in Figure 8 has only three nodes by redrawing the circuit in Figure 9. The two circuits in Figure 8 and Figure 9 are identical. However, for the sake of the clarity, nodes b and c are spread out with perfect conductors as in Figure 8.

A loop is a closed path formed by starting at a node, passing through any node more than one. A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop. Independent loops or paths result in independent sets of equations.

For example, the closed path abca containing the 2- ΩΩ size 12{ %OMEGA } {} resistor in Figure 9 is a loop. Another loop is the closed path bcb containing the 3- ΩΩ size 12{ %OMEGA } {} resistor and current source. Although one can identify six loops in Figure 9, only three of them are independent.

A network with b branches, n nodes, and l independent loops will satisfy the fundamental theorem of network topology:

b = l + n - 1

As the next two definitions show, circuit topology is of great value to the study of voltages and currents in an electric circuit.

Elements are in series when they are chain-connected or connected sequentially, end to end. For example, two elements are in series if they share one common node and no other element is connected to that common node. Elements in parallel are connected to the same pair of terminals. Elements may be connected in a way that they are neither in series nor in parallel. In the circuit show in Figure 8, the voltage source and the 5- ΩΩ size 12{ %OMEGA } {} resistor are in series because the same current will flow through them. The 2- ΩΩ size 12{ %OMEGA } {} resistor, the 3- ΩΩ size 12{ %OMEGA } {} resistor, and the current source are in parallel because they are connected to the same two nodes (b and c) resistors and neither in series nor in parallel with each other.

Ohm’s law is not sufficient to analyze circuits. However, when it is coupled with Kirchhoff’s two laws, we have a sufficient, powerful set of tools for analyzing a large variety of electric circuits. Kirchhoff’s laws were first introduced in 1847 by the German physicist Gustav Robet Kirchhoff (1824-1887). These laws are formally known as Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL).

Kirchhoff’s first law is based on the law of conservation of charge, which requires that the algebraic sum of charges within a system cannot change.

Mathematically, KCL implies that

Where N is number of branches connected to the node and in is nth current entering (or leaving) the node. By the law, current entering a node may be regarded as positive, while currents leaving the node may be taken as negative or vice versa.

To prove KCL assume a set of currents ik(t)ik(t) size 12{i rSub { size 8{k} } \( t \) } {}, k = 1, 2 … flow into a node. The algebraic sum of currents at the node is

Integrating both sides of Equation 13 gives

Where qk(t)=∫ik(t)dtqk(t)=∫ik(t)dt size 12{q rSub { size 8{k} } \( t \) = Int {i rSub { size 8{k} } \( t \) ital "dt"} } {}and qT(t)=∫iT(t)dtqT(t)=∫iT(t)dt size 12{q rSub { size 8{T} } \( t \) = Int {i rSub { size 8{T} } \( t \) ital "dt"} } {} but the law of conservation of electric charge requires that the algebraic sum of electric charges at the note must not change; that is, the node stores no net charge. Thus qT(t)=0→i(t)=0qT(t)=0→i(t)=0 size 12{q rSub { size 8{T} } \( t \) =0 rightarrow i \( t \) =0} {}, confirming the validity of KCL.

Consider the node in Figure 10. Applying KCL gives

Since current i1i1 size 12{i rSub { size 8{1} } } {}, i2i2 size 12{i rSub { size 8{2} } } {} and i4i4 size 12{i rSub { size 8{4} } } {} are entering the node, while current i2i2 size 12{i rSub { size 8{2} } } {}and i5i5 size 12{i rSub { size 8{5} } } {} are leaving it. By rearranging the terms, we get

Equation 16 is an alternative form of KCL:

Note that KCL also applies to a closed boundary. This may be regarded as generalized case, because a node may be regarded as a closed surface shrunk to a point. In two dimensions, a closed boundary is the same as a closed path. As typically illustrated in the circuit of Figure 11, the total current entering the closed surface is equal to the total current leaving the surface.

A simple application of KCL is combining current sources in parallel. The combined current is the algebraic sum of the current supplied by the individual sources. For examples, the current sources shown in Figure 12(a) can be combined as in Figure 12(b). The combined or equivalent current source can be found by applying KCL to node a.

I T + I 2 = I 1 + I 3 I T + I 2 = I 1 + I 3 size 12{I rSub { size 8{T} } +I rSub { size 8{2} } =I rSub { size 8{1} } +I rSub { size 8{3} } } {}

or

A circuit cannot contain two different currents, I1I1 size 12{I rSub { size 8{1} } } {} and I2I2 size 12{I rSub { size 8{2} } } {}, in series, unless I1=I2I1=I2 size 12{I rSub { size 8{1} } =I rSub { size 8{2} } } {}; otherwise KCL will be violated.

Kirchhoff’s second law is based on the principle of conservation of energy:

Expressed mathematically, KVL states that

Where M is the number of voltages in the loop (or the number of branches in the loop) and vm is the mth voltage.

To illustrate KVL, consider the circuit in Figure 13. The sign on each voltage is the polarity of the terminal encountered first as we travel around the loop. We can start with any branch and go around the loop either clockwise or counterclockwise. Suppose we start with the voltage source and go clockwise around the loop as shown; then voltages would be −v1−v1 size 12{ - v rSub { size 8{1} } } {}, +v2+v2 size 12{+v rSub { size 8{2} } } {}, +v3+v3 size 12{+v rSub { size 8{3} } } {}, −v4−v4 size 12{ - v rSub { size 8{4} } } {}, and +v5+v5 size 12{+v rSub { size 8{5} } } {}, in that order. For example, as we reach branch 3, the positive terminal is met first; hence we have +v3+v3 size 12{+v rSub { size 8{3} } } {}. For branch 4, we reach the negative terminal first; hence, −v4−v4 size 12{ - v rSub { size 8{4} } } {}. Thus, KVL yields

Rearranging terms gives