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Chapter 2 Basic Circuit Analysis 1353T_c02_014-065.qxd 7/28/05 10:56 AM Page 14 CHAPTER 2 BASIC CIRCUIT ANALYSIS The equation S ϭ A/L shows that the current of a voltaic circuit is subject to a change, by each variation originating either in the mag- nitude of a tension or in the reduced length of a part, which latter is itself again determined, both by the actual length of the part as well as its conductivity and its section. Georg Simon Ohm, 1827, German Mathematician/Physicist Some History Behind This Chapter Chapter Learning Objectives Georg Simon Ohm (1789–1854) discovered the law that 2-1 Element Constraints (Sect. 2-1) now bears his name in 1827. His results drew heavy crit- Given a two-terminal element with one or more electrical icism and were not generally accepted for many years. variables specified, use the element iϪv constraint to find Fortunately, the importance of his contribution was even- the magnitude and direction of the unknown variables. tually recognized during his lifetime. He was honored by the Royal Society of England in 1841 and appointed a Pro- 2-2 Connection Constraints (Sect. 2-2) fessor of Physics at the University of Munich in 1849. Given a circuit composed of two-terminal elements: (a) Identify nodes and loops in the circuit. Why This Chapter Is Important Today (b) Identify elements connected in series and in parallel. A circuit is an interconnection of electric devices that per- (c) Use Kirchhoff’s laws (KCL and KVL) to find forms a useful function. This chapter introduces some basic selected signal variables. tools you will need to analyze and design electric circuits. You will also be introduced to several important electric 2-3 Combined Constraints (Sect. 2-3) devices that control currents and voltages in a circuit. These Given a linear resistance circuit, use the element con- devices range from everyday things like batteries to spe- straints and connection constraints to find selected signal cial integrated circuits that meter out predetermined volt- variables. ages or currents. 2-4 Equivalent Circuits (Sect. 2-4) Chapter Sections Given a linear resistance circuit, find an equivalent circuit 2–1 Element Constraints at a specified pair of terminals. 2–2 Connection Constraints 2-5 2–3 Combined Constraints Voltage and Current Division (Sect. 2-5) 2–4 Equivalent Circuits (a) Given a linear resistance circuit with elements con- nected in series or in parallel, use voltage or current 2–5 Voltage and Current Division division to find specified voltages or currents. 2–6 Circuit Reduction (b) Design a voltage or current divider that delivers 2–7 Computer-Aided Circuit Analysis specified output signals. 2-6 Circuit Reduction (Sect. 2-6) Given a linear resistance circuit, find selected signal vari- ables using successive application of series and parallel equivalence, source transformations, and voltage and cur- rent division. 1353T_c02_014-065.qxd 7/28/05 10:56 AM Page 15 ELEMENT CONSTRAINTS S ECTION 2–1 15 2–1 E LEMENT C ONSTRAINTS A circuit is a collection of interconnected electrical devices. An electrical de- vice is a component that is treated as a separate entity. The rectangular box in i Figure 2–1 is used to represent any one of the two-terminal devices used to + form circuits. A two-terminal device is described by its i–v characteristic; that is, by the relationship between the voltage across and current through the v device. In most cases the relationship is complicated and nonlinear, so we use Device a linear model that approximates the dominant features of a device. − To distinguish between a device (the real thing) and its model (an approxi- FIGURE 2–1 Voltage and mate stand-in), we call the model a circuit element. Thus, a device is an article current reference marks for a two- of hardware described in manufacturers’ catalogs and parts specifications. An terminal device. element is a model described in textbooks on circuit analysis. This book is no exception, and a catalog of circuit elements will be introduced as we go on. A discussion of real devices and their models is contained in Appendix A. i + THE LINEAR RESISTOR vR The first element in our catalog is a linear model of the device described in − Figure 2–2. The actual i–v characteristic of this device is shown in Figure 2–2(b). To model this curve accurately across the full operating range shown (a) in the figure would require at least a cubic equation. However, the graph in i Figure 2–2(b) shows that a straight line is a good approximation to the i–v Actual characteristic if we operate the device within its linear range. The power rat- ing of the device limits the range over which the i–v characteristics can be Model represented by a straight line through the origin. Linear range 1 For the passive sign convention used in Figure 2–2(a), the equations de- scribing the linear resistor element are R v v ϭ Ri or i ϭ Gv (2–1) where R and G are positive constants that are reciprocally related. 1 G ϭ (2–2) R Power rating limits The relationships in Eq. (2–1) are collectively known as Ohm’s law. The para- (b) meter R is called resistance and has the unit ohms, ⍀. The parameter G is called conductance, with the unit siemens, S. In earlier times the unit of con- ductance was cleverly called the mho, with a unit abbreviation symbol ⍀ (ohm spelled backward and the ohm symbol upside down). Note that Ohm’s law presumes that the passive sign convention is used to assign the reference marks to voltage and current. The Ohm’s law model is represented graphically by the black straight Wirewound line in Figure 2–2(b). The i–v characteristic for the Ohm’s law model defines a circuit element that is said to be linear and bilateral. Linear means that the defining characteristic is a straight line through the origin. Elements whose characteristics do not pass through the origin or are not a straight line are said to be nonlinear. Bilateral means that the i–v characteristic curve has odd symmetry about the origin.1 With a bilateral resistor, revers- Carbon or film ing the polarity of the applied voltage reverses the direction but not the (c) FIGURE 2–2 The resistor: (a) Circuit symbol. (b) i–␷ character- 1A curve i = f(v) has odd symmetry if f(−v) = −f(v). istics. (c) Some actual devices. 1353T_c02_014-065.qxd 08:23:2005 3:47 PM Page 16 16 C HAPTER 2 BASIC CIRCUIT ANALYSIS magnitude of the current, and vice versa. The net result is that we can con- nect a bilateral resistor into a circuit without regard to which terminal is which. This is important because devices such as diodes and batteries are not bilateral, and we must carefully identify each terminal. Figure 2–2(c) shows sketches of discrete resistor devices. Detailed device characteristics and fabrication techniques are discussed in Appendix A. The power associated with the resistor can be found from p = vi. Using Eq. (2–1) to eliminate v from this relationship yields p ϭ i2R (2–3) ii or using the same equations to eliminate i yields + + v2 p ϭ v2G ϭ (2–4) R v Openv Short Since the parameter R is positive, these equations tell us that the power is − − always nonnegative. Under the passive sign convention, this means that the resistor always absorbs power. (a)(b) FIGURE 2–3 Circuit sym- EXAMPLE 2–1 bols: (a) Open-circuit symbol. (b) Short-circuit symbol. A resistor operates as a linear element as long as the voltage and current are within the limits defined by its power rating. Suppose we have a 47-k⍀ resistor with a power rating of 0.25 W. Determine the maximum current and voltage that can be applied to the resistor and remain within its linear i operating range. SOLUTION: i Using Eq. (2–3) to relate power and current, we obtain + OFF v P (open) ϭ MAX ϭ 0.25 ϭ I ͱ ͱ 2.31 mA v MAX R 47 ϫ 103 − Similarly, using Eq. (2–4) to relate power and voltage, we obtain Circuit symbol i-v characteristics ϭ ͙ ϭ ͙ ϫ 3 ϫ ϭ ■ (a) VMAX RPMAX 47 10 0.25 108 V PEN AND SHORT CIRCUITS i O The next two circuit elements can be thought of as limiting cases of the lin- ear resistor. Consider a resistor R with a voltage v applied across it. Let’s i calculate the current i through the resistor for different values of resis- tance. If v = 10 V and R = 1 ⍀, using Ohm’s law we readily find that i = + ON v ⍀ (closed) 10 A. If we increase the resistance to 100 , we find i has decreased to v 0.1 A or 100 mA. If we continue to increase R to 1 M⍀, i becomes a very ␮ − small 10 A. Continuing this process, we arrive at a condition where R is very nearly infinite and i just about zero. When the current i = 0, we call ∞ ⍀ Circuit symbol i-v characteristics the special value of resistance (i.e., R = ) an open circuit. Similarly, if (b) we reduce R until it approaches zero, we find that the voltage is very nearly zero. When v = 0, we call the special value of resistance (i.e., R = FIGURE 2–4 The circuit ⍀ symbol and i–v characteristics of 0 ), a short circuit. The circuit symbols for these two elements are shown an ideal switch: (a) Switch OFF.
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