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chapter 10 10.1 A N A L Y S I S O F R C C I R C U I T S 10.2 A N A L Y S I S O F R L C I R C U I T S 10.3 I N T U I T I V E A N A L Y S I S 10.4 P R O P A G A T I O N D E L A Y A N D T H E D I G I T A L A B S T R A C T I O N 10.5 S T A T E A N D S T A T E V A R I A B L E S 10.6 A D D I T I O N A L E X A M P L E S 10.7 D I G I T A L M E M O R Y 10.8 S U M M A R Y E X E R C I S E S P R O B L E M S first-order transients in linear electrical networks 10 AsillustratedinChapter9, capacitancesandinductancesimpactcircuitbehavior. The effect of capacitances and inductances is so acute in high-speed digital cir-cuits, for example, that our simple digital abstractions developed in Chapter 6 based on a static discipline become insufficient for signals that undergo transi-tions. Therefore, understanding the behavior of circuits containing capacitors and inductors is important. In particular, this chapter will augment our digital abstraction with the concept of delay to include the effects of capacitors and inductors. Looked at positively, because they can store energy, capacitors and induc-tors display the memory property, and offer signal-processing possibilities not available in circuits containing only resistors. Apply a square-wave voltage to a multi-resistor linear circuit, and all of the voltages and currents in the network will have the same square-wave shape. But include one capacitor in the circuit and very different waveforms will appear sections of exponentials, spikes, and sawtooth waves. Figure 10.1 shows an example of such waveforms for the two-inverter system of Figure 9.1 in Chapter 9. The linear analysis tech-niquesalreadydeveloped nodeequations, superposition, etc. areadequate for finding appropriate network equations to analyze these kinds of circuits. However, the formulations turn out to be differential equations rather than algebraic equations, so additional skills are needed to complete the analyses. vI vO vI t t . . . FIGURE 10.1 Observed response of the first inverter + + to a square-wave input. - v - 503 504 C H A P T E R T E N f i r s t - o r d e r t r a n s i e n t s This chapter will discuss systems containing a single storage element, namely, a single capacitor or a single inductor. Such systems are described by simple, first-order differential equations. Chapter 12 will discuss systems containing two storage elements. Systems with two storage elements are described by second-order differential equations.1 Higher-order systems are also possible, and are discussed briefly in Chapter 12. Thischapterwillstartbyanalyzingsimplecircuitscontainingonecapacitor, one resistor, and possibly a source. We will then analyze circuits containing one inductor and one resistor. The two-inverter circuit of Figure 10.1 is examined in detail in Section 10.4. 10.1 ANALYSIS OF RC CIRCUITS Let us illustrate first-order systems with a few primitive examples containing a resistor, a capacitor, and a source. We first analyze a current source driving the so-called parallel RC circuit. 10.1.1 PARALLEL RC CIRCUIT, STEP INPUT Shown in Figure 10.2a is a simple source-resistor-capacitor circuit. On the basis of the Thévenin and Norton equivalence discussion in Section 3.6.1, this circuit could result from a Norton transformation applied to a more complicated (a) i(t) R C +v (t) i(t) I0 FIGURE 10.2 Capacitor (b) charging transient. 0 t v I0R (c) Time constant RC 0 t 1. However, a circuit with two storage elements that can be replaced by a single equivalent storage element remains a first-order circuit. For example, a pair of capacitors in parallel can be replaced with a single capacitor whose capacitance is the sum of the two capacitances. 10.1 Analysis of RC Circuits C H A P T E R T E N 505 R1 R3 v1 + i1 R2 + - v2 vC R4 C - R5 i2 FIGURE 10.3 A more complicated circuit that can be transformed into the simpler circuit in Figure 10.2a by using Thévenin and Norton transformations. i R C +vC circuit containing many sources and resistors, and one capacitor, as suggested in Figure 10.3. Let us assume we wish to find the capacitor voltage vC. We will usethenodemethoddescribedinChapter3todoso. AsshowninFigure10.2a, we take the bottom node as ground, which leaves us with one unknown node voltage corresponding to the top node. The voltage at the top node is the same as the voltage across the capacitor, and so we will proceed to work with vC as our unknown. Next, according to Step 3 of the node method, we write KCL for the top node in Figure 10.2a, substituting the constituent relation for a capacitor from Equation 9.9, i(t) = R +C dt . (10.1) Or, rewriting, dvC vC i(t) dt RC C (10.2) As promised, the problem can be formulated in one line. But to find vC(t), we must solve a nonhomogeneous, linear first-order ordinary differential equation with constant coefficients. This is not a difficult task, but one that must be done systematically using any method of solving differential equations. To solve this equation, we will use the method of homogeneous and par-ticular solutions because this method can be readily extended to higher-order equations. As a review, the method of homogeneous and particular solutions arises from a fundamental theorem of differential equations. The method states that the solution to the nonhomogeneous differential equation can be obtained by summing together the homogeneous solution and the particular solution. More specifically, let vCH(t) be any solution to the homogeneous differential ... - tailieumienphi.vn
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