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Methods of Analysis

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Methods of Analysis

No great work is ever done in a hurry. To develop a great scientific discovery, to print a great picture, to write an immortal poem, to become a minister, or a famous general—to do anything great requires time, patience, and perseverance. These things are done by degrees, “little by little.”

—W. J. Wilmont Buxton

Enhancing Your Career

Career in Electronics

One area of application for electric circuit analysis is electronics. The term electronics was originally used to distinguish circuits of very low current levels. This distinction no longer holds, as power semiconductor devices operate at high le vels of current. Today, electronics is regarded as the science of the motion of char ges in a g as, vacuum, or semicon ductor. Modern electronics in volves transistors and transistor circuits. The earlier electronic circuits were assembled from components. Man y electronic circuits are now produced as integrated circuits, fabricated in a semiconductor substrate or chip.

Electronic circuits find applications in many areas, such as automation, broadcasting, computers, and instrumentation. The range of devices that use electronic circuits is enormous and is limited only by our imagination. Radio, television, computers, and stereo systems are but a few.

An electrical engineer usually performs di verse functions and is likely to use, design, or construct systems that incorporate some form of electronic circuits. Therefore, an understanding of the operation and analysis of electronics is essential to the electrical engineer . Electronics has become a specialty distinct from other disciplines within electrical engi neering. Because the field of electronics is ever advancing, an electronics engineer must update his/her knowledge from time to time. The best way to do this is by being a member of a professional organization such as the Institute of Electrical and Electronics Engineers (IEEE). With a membership of over 300,000, the IEEE is the largest professional organization in the world. Members benefit immensely from the numerous magazines, journals, transactions, and conference/symposium proceedings published yearly by IEEE. You should consider becoming an IEEE member.

Troubleshooting an electronic circuit board. © Brand X Pictures/PunchStock RF

Learning Objectives

By using the information and exercises in this chapter you will be able to:

    1. Understand Kirchhoff’s current law.
    1. Understand Kirchhoff’s voltage law.
    1. Develop an understanding of how to use Kirchhoff’s current law to write nodal equations and then to solve for unknown node voltages.
    1. Develop an understanding of how to use Kirchhoff’s voltage law to write mesh equations and then to solve for unknown loop currents.
    1. Explain how to use PSpice to solve for unknown node voltages and currents.

3.1 Introduction

Having understood the fundamental laws of circuit theory (Ohm’s law and Kirchhoff’s laws), we are now prepared to apply these laws to develop two powerful techniques for circuit analysis: nodal analysis, which is based on a systematic application of Kirchhof f’s current law (KCL), and mesh analysis, which is based on a systematic application of Kirchhoff’s voltage law (KVL). The two techniques are so important that this chapter should be re garded as the most important in the book. Students are therefore encouraged to pay careful attention.

With the two techniques to be developed in this chapter, we can analyze any linear circuit by obtaining a set of simultaneous equations that are then solved to obtain the required values of current or voltage. One method of solving simultaneous equations involves Cramer’s rule, which allows us to calculate circuit variables as a quotient of determinants. The examples in the chapter will illustrate this method; Appendix A also briefly summarizes the essentials the reader needs to kno w for applying Cramer’ s rule. Another method of solving simultaneous equations is to use MATLAB, a computer software discussed in Appendix E.

Also in this chapter, we introduce the use of PSpice for Windows, a circuit simulation computer software program that we will use throughout the text. Finally, we apply the techniques learned in this chapter to analyze transistor circuits.

3.2 Nodal Analysis

Nodal analysis provides a general procedure for analyzing circuits using node voltages as the circuit v ariables. Choosing node v oltages instead of element voltages as circuit variables is convenient and reduces the number of equations one must solve simultaneously.

To simplify matters, we shall assume in this section that circuits do not contain voltage sources. Circuits that contain voltage sources will be analyzed in the next section.

In nodal analysis, we are interested in finding the node voltages. Given a circuit with n nodes without voltage sources, the nodal analysis of the circuit involves taking the following three steps.

Nodal analysis is also known as the node-voltage method.

Steps to Determine Node Voltages:

    1. Select a node as the reference node. Assign voltages v1, v2, … , vn− 1 to the remaining n − 1 nodes. The voltages are referenced with respect to the reference node.
    1. Apply KCL to each of the n − 1 nonreference nodes. Use Ohm’s law to express the branch currents in terms of node voltages.
    1. Solve the resulting simultaneous equations to obtain the un known node voltages.

We shall now explain and apply these three steps.

The first step in nodal analysis is selecting a node as the reference or datum node. The reference node is commonly called the ground since it is assumed to have zero potential. A reference node is indicated b y any of the three symbols in Fig. 3.1. The type of ground in Fig. 3.1(c) is called a chassis ground and is used in de vices where the case, enclosure, or chassis acts as a reference point for all circuits. When the potential of the earth is used as reference, we use the earth ground in Fig. 3.1(a) or (b). We shall always use the symbol in Fig. 3.1(b).

Once we ha ve selected a reference node, we assign v oltage designations to nonreference nodes. Consider , for e xample, the circuit in Fig. 3.2(a). Node 0 is the reference node ( v = 0), while nodes 1 and 2 are assigned voltages v1 and v2, respectively. Keep in mind that the node voltages are defined with respect to the reference node. As illustrated in Fig. 3.2(a), each node voltage is the voltage rise from the reference node to the corresponding nonreference node or simply the v oltage of that node with respect to the reference node.

As the second step, we apply KCL to each nonreference node in the circuit. To avoid putting too much information on the same circuit, the circuit in Fig. 3.2(a) is redra wn in Fig. 3.2(b), where we no w add i1, i2, and i3 as the currents through resistors R1, R2, and R3, respectively. At node 1, applying KCL gives

I1=I2+i1+i2(3.1)I_1 = I_2 + i_1 + i_2 \tag{3.1}

At node 2,

I2+i2=i3(3.2)I_2 + i_2 = i_3 \tag{3.2}

We now apply Ohm’s law to express the unknown currents i1, i2, and i3 in terms of node voltages. The key idea to bear in mind is that, since resistance is a passive element, by the passive sign convention, current must always flow from a higher potential to a lower potential.

Current flows from a higher potential to a lower potential in a resistor.

We can express this principle as

i=vhigher−vlowerRi = \frac{v_{\text{higher}} - v_{\text{lower}}}{R}

(3.3)

Figure 3.1