Comprehensive Problems

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Comprehensive Problems

7.88 The circuit in Fig. 7.148(a) can be designed as an approximate differentiator or an integrator, depending on whether the output is taken across the resistor or the capacitor, and also on the time constant τ = RC of the circuit and the width T of the input pulse in Fig. 7.148(b). The circuit is a differentiator if τ≪ T, say τ< 0.1T, or an integrator if τ≫ T, say τ> 10T.
(a) What is the minimum pulse width that will allow a differentiator output to appear across the capacitor?
(b) If the output is to be an integrated form of the input, what is the maximum value the pulse width can assume?

For Prob. 7.88.

7.89 An RL circuit may be used as a differentiator if the output is taken across the inductor and τ≪ T (say τ< 0.1T), where T is the width of the input pulse. If R is fixed at 200 kΩ, determine the maximum value of L required to differentiate a pulse with T = 10 µs.

7.90 An attenuator probe employed with oscilloscopes was designed to reduce the magnitude of the input voltage vi by a factor of 10. As shown in Fig. 7.149, the oscilloscope has internal resistance Rs and capacitance Cs, while the probe has an internal resistance Rp. If Rp is fixed at 6 MΩ, find Rs and Cs for the circuit to have a time constant of 15 µs.

Figure 7.149 For Prob. 7.90.

7.91 The circuit in Fig. 7.150 is used by a biology student to study “frog kick.” She noticed that the frog kicked a little when the switch was closed but kicked violently for 5 s when the switch was opened. Model the frog as a resistor and calculate its resistance. Assume that it takes 10 mA for the frog to kick violently.

Figure 7.150 For Prob. 7.91.

7.92 To move a spot of a cathode-ray tube across the screen requires a linear increase in the voltage across the deflection plates, as shown in Fig. 7.151. Given that the capacitance of the plates is 4 nF, sketch the current flowing through the plates.

Figure 7.151 For Prob. 7.92.

chapter

Second-Order Circuits

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Enhancing Your Career

To increase your engineering career opportunities after graduation, develop a strong fundamental understanding in a broad set of engineer ing areas. When possible, this might best be accomplished by working toward a graduate degree immediately upon receiving your undergraduate degree.

Each de gree in engineering represents certain skills the student acquires. At the Bachelor de gree level, you learn the language of engi neering and the fundamentals of engineering and design. At the Master’s level, you acquire the ability to do advanced engineering projects and to communicate your work effectively both orally and in writing. The Ph.D. represents a thorough understanding of the fundamentals of electrical engineering and a mastery of the skills necessary both for w orking at the frontiers of an engineering area and for communicating one’ s effort to others.

If you have no idea what career you should pursue after graduation, a graduate de gree program will enhance your ability to e xplore career options. Since your undergraduate degree will only provide you with the fundamentals of engineering, a Master’ s degree in engineering supple mented by business courses benefits more engineering students than does getting a Master’s of Business Administration (MBA). The best time to get your MB A is after you ha ve been a practicing engineer for some years and decide your career path would be enhanced by strengthening your business skills.

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Enhancing your career involves understanding your goals, adapting to changes, anticipating opportunities, and planning your own niche.

Learning Objectives

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

1. Develop a better understanding of the solution of generalsecond order differential equations.
1. Learn how to determine initial and final values.
1. Understand the response in source-free series RLC circuits.
1. Understand the response in source-free parallel RLC circuits.
1. Understand the step response of series RLC circuits.
1. Understand the step response of parallel RLC circuits.
1. Understand general second-order circuits.
1. Understand general second-order circuits with op amps.

8.1 Introduction

In the previous chapter we considered circuits with a single storage element (a capacitor or an inductor). Such circuits are first-order because the differential equations describing them are first-order. In this chap ter we will consider circuits containing two storage elements. These are known as second-order circuits because their responses are described by differential equations that contain second derivatives.

Typical e xamples of second-order circuits are RLC circuits, in which the three kinds of passive elements are present. Examples of such circuits are shown in Fig. 8.1(a) and (b). Other examples are RL and RC circuits, as shown in Fig. 8.1(c) and (d). It is apparent from Fig. 8.1 that a second-order circuit may have two storage elements of different type or the same type (provided elements of the same type cannot be represented by an equi valent single element). An op amp circuit with tw o storage elements may also be a second-order circuit. As with first-order circuits, a second-order circuit may contain se veral resistors and dependent and independent sources.

A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements.

Our analysis of second-order circuits will be similar to that used for first-order. We will first consider circuits that are excited by the initial conditions of the storage elements. Although these circuits may contain dependent sources, they are free of independent sources. These sourcefree circuits will give natural responses as expected. Later we will consider circuits that are e xcited by independent sources. These circuits will give both the transient response and the steady-state response. We consider only dc independent sources in this chapter. The case of sinusoidal and exponential sources is deferred to later chapters.

We begin by learning ho w to obtain the initial conditions for the circuit variables and their deri vatives, as this is crucial to analyzing second-order circuits. Then we consider series and parallel RLC circuits such as shown in Fig. 8.1 for the two cases of excitation: by initial