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 comprehensive treatment of two terminal resistors However unlike the usual terminology a resistor may be linear nonlinear time invariant or timevarying 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 characterized 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 t o dc operating points and the derivation of the smallsignal 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 Circuits 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 about the 45 line the dual characteristic is obtained by reflecting through the origin Power passive resistors active resistors and modeting The symbol for a two terminal nonlinear resistor 2 is shown in Fig 1 6 The power delivered ro the resistor at time t by the remainder of the circuit to which it is connected is from Chap 1 If the resistor is linear having resistance R Thus the power delivered to a linear resistor is always nonnegative if R 0 We say that a linear resistor is passive iff its resistance is nonnegative Thus a passive resistor always absorbs energy from
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