CHAPTER TWO TWO-TERMINAL RESISTORS Two-terminal elements play a major role in electric circuits. As a matter of fact, many introductory texts on electric circuits consider circuits consisting only of two-terminal elements exclusively. In this chapter we give a com- prehensive treatment of two-terminal resistors. However. unlike the usual terminology, a resistor may be linear, nonlinear. time-invariant. or time- varying. It is characterized by a relation between the branch voltage and the branch current. We speak of the v-i characteristic of a resistor, and we discuss the characteristics of various types of resistors such as a linear resistor ~vhich satisfies Ohm's law, an ideal diode, a dc current source. a pn-junction diode. and a periodically operating switch. All of these are resistors. By interconnecting two-terminal resistors, we form a resistive circuit. The simplest forms of interconnection, i.e., series, parallel. and series-parallel interconnections, will be treated and illustrated with examples. These require the use of Kirchhoff's laws together with branch equations which characterize the elements. A one-port formed by the interconnection of resistors is charact- erized by its driving-point characteristics relating its port voltage and its port current. We introduce the concepts of equivalence and duality of one-ports by simple examples. These will be generalized in later chapters. An important problem in nonlinear circuits is the determination of the dc operating points, i.e., the solutions with dc inputs. Various methods and techniques are introduced and illustrated. Another important problem in nonlinear circuits is the small-signal analysis. Its relation to dc operating points and the derivation of the small- signal equivalent circuit are treated by way of a simple example. This subject will be discussed in a more general fashion in later chapters.46 LlNEAR AND NONLINEAR CIRCUITS Finally. we discuss the transfer characteristic of resistive circuits and demonstrate the usefulness of the graphic method in analyzing nonlinear resistive circuits. 1 v-i CHARACTERISTIC OF TWO-TERMINAL RESISTORS 1.1 From Linear Resistor to Resistor The most familiar circuit element that one encounters in physics or in an elementary electrical engineering course is a two-terminal resistor which satisfies Ohm's law: i.e.. the voltage across such an element is proportional to the current flowing through it. We call such an element a linear resistor. We represent it by the symbol shown in Fig. 1.1, where the current i through the resistor and the voltage v across ir are measured using the associated reference directions. Ohm's law states that, at all times where the constant R is the resistance of the linear resistor measured in the unit of ohms (R), and G is the condzictance measured in the unit of siemens (S). The voltage u(t) and the current i(t) in Eq. (1 .l) are expressed in volts (V) and amperes (A). respectively. Equation (1.1) can be plotted on the i-v plane or the v-i plane' as shown in Fig. 1.2~ and b. where the slope in each is the resistance and the conductance, respectively. While the linear resistor is perhaps the most prevalent circuit element in electrical engineering, nonlinear devices which can be modeled with nonlinear resistors have become increasingly important. Thus it is necessary to define the concept of nonlinear resistor in a most general way. Consider a two-terminal element as shown in Fig. 1.3. The voltage v across the element and the current i which enters the element through one terminal and leaves from the other are shown using the associated reference directions. A two-terminal element will be called a resistor if its voltage v and current i - Figure 1.1 Symbol for a linear resistor with resistance R. ' When we say X-y plane, we denote specifically x as the horizontal axis and y as the vertical axis of the plane. This is consistent with the conventional usage where the first variable denotes the abscissa and the second variable denotes the ordinate.TWO-TERMINAL RESISTORS 47 Figure 1.2 Linear resistor characteristic plotted (a) on the i-v plane and (6) on the v-i plane. v Figure 1.3 A two-terminal element with v and i in the associated reference directions. satisfy the following relation: iRR = {(U, i): f(v, i) = 0) This relation is called the v-i characteristic of the resistor and can be plotted graphically in the v-i plane (or i-v plane). The equation f(v, i) = 0 represents a curve in the v-i plane (or i-v plane) and specifies completely the two-terminal resistor. The key idea of a resistor is that in Eq. (1.2) the relation is between v(t), the instantaneous value of the voltage v(-) and i(t) the instantaneous value of the current i(.) at time t. The dc voltage versus current characteristics of devices can be measured using a curve tracer? The linear resistor is a special case of a resistor in which A resistor which is not linear is called nonlinear. Before considering nonlinear resistors, we should first understand linear resistors. Equations (1.1) and (1.3) state that, for a linear resistor, the relation between the voltage v and current i is expressed by linear functions. The first equation in (1.1) expresses v as a linear function of i, and the second equation in (1.1) expresses i as a linear function of v. Figure 1.2 shows that the v-i See for example: J. Mulvey, Semiconductor Device Mearurements, Tektronix Inc., Beavenon, Oregon, 1968. L. 0. Chua and G. Q. Zhong, "Negative Resistance Curve Tracer," IEEE Transactions on Cir- cuits and System, vol. CAS-32, pp. 569-582, June 1985.TWO-TERMINAL RESISTORS 49 curve as that of the given resistor in the i-v plane. The concept of duality is of utmost importance in circuit theory. It helps us in understanding and analyzing circuits of great generality. We will encounter duality throughout this book. Exercises 1. A linear resistor of 100 L! is given; what is its dual? 2. If 2, = {(v. i): f(v, i) = v - i3 = 0) specifies a resistor. write down the relation of the dual resistor. 3. Given the v-i characteristic r of a resistor 9 on the v-i plane, show that the dual characteristic is obtained by reflecting about the 45" line through the origin. Power, passive resistors, active resistors, and modeting
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