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544 Find i for t > 0 in the circuit of Figure P544 if
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i(0) = 4 A and v(0) = 6 V
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540 Assume the circuit of Figure P540 initially stores
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no energy The switch is closed at t = 0, and then reopened at t = 50 s Determine an expression for the capacitor voltage for t 0
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545 Find v for t > 0 in the circuit of Figure P545 if the
Figure P540
circuit is in steady state at t = 0
Part I
Circuits
4 12 V + _
1H + v 1/4 F
3 +
20 A
1/6 F
1/6 F
Figure P545
Figure P548
546 Find i for t > 0 in the circuit of Figure P546 if the
circuit is in steady state at t = 0
2 40 V + _ 1H t=0 2 3
549 The circuit of Figure P549 is in steady state at
1H i 2
+ _ 10 V
t = 0 Find v for t > 0 if L is (a) 24 H, (b) 3 H, and (c) 4 H
3 t=0 1/12 F + v L
Figure P546
547 Find i for t > 0 in the circuit of Figure P547 if the
circuit is in steady state at t = 0
2 40 V + _ 1H t=0 i 2 3
Figure P549
550 Find v for t > 0 in the circuit of Figure P550 if the
circuit is in steady state at t = 0
2 + v 4V + _ 1/4 F
+ 12 V 08 H _
t=0 3
Figure P547
548 Find v for t > 0 in the circuit of Figure P548 if the
circuit is in steady state at t = 0
Figure P550
Frequency Response and System Concepts
hapter 4 introduced the notions of energy-storage elements and dynamic circuit equations and developed appropriate tools (complex algebra and phasors) for the solution of AC circuits In 5, we explored the solution of rst- and second-order circuits subject to switching transients The aim of the present chapter is to exploit AC circuit analysis methods to study the frequency response of electric circuits It is common, in engineering problems, to encounter phenomena that are frequency-dependent For example, structures vibrate at a characteristic frequency when excited by wind forces (some high-rise buildings experience perceptible oscillation!) The propeller on a ship excites the shaft at a vibration frequency related to the engine s speed of rotation and to the number of blades on the propeller An internal combustion engine is excited periodically by the combustion events in the individual cylinder, at a frequency determined by the ring of the cylinders Wind blowing across a pipe excites a resonant vibration that is perceived as sound (wind instruments operate on this principle) Electrical circuits are no different from other dynamic systems in this respect, and a large body of knowledge has been developed for understanding the frequency response of electrical circuits, mostly based on the ideas behind phasors and impedance These ideas, and the concept of ltering, will be explored in this chapter The ideas developed in this chapter will also be applied, by analogy, to the analysis of other physical systems (eg, mechanical systems), to illustrate the generality of the concepts By the end of the chapter, you should be able to:
6
Frequency Response and System Concepts
Compute the frequency response function for an arbitrary circuit Use knowledge of the frequency response to determine the output of a circuit Recognize the analogy between electrical circuits and other dynamic systems
SINUSOIDAL FREQUENCY RESPONSE
The sinusoidal frequency response (or, simply, frequency response) of a circuit provides a measure of how the circuit responds to sinusoidal inputs of arbitrary frequency In other words, given the input signal amplitude, phase, and frequency, knowledge of the frequency response of a circuit permits the computation of the output signal The box Fourier Analysis provides further explanation of the importance of sinusoidal signals Suppose, for example, that you wanted to determine how the load voltage or current varied in response to different excitation signal frequencies in the circuit of Figure 61 An analogy could be made, for example, with how a speaker (the load) responds to the audio signal generated by a CD player (the source) when an ampli er (the circuit) is placed between the two1 In the circuit of Figure 61, the signal source circuitry is represented by its Th venin equivalent Recall that the impedance ZS presented by the source to the e remainder of the circuit is a function of the frequency of the source signal (Section 44) For the purpose of illustration, the ampli er circuit is represented by the idealized connection of two impedances, Z1 and Z2 , and the load is represented by an additional impedance, ZL What, then, is the frequency response of this circuit The following is a fairly general de nition:
The frequency response of a circuit is a measure of the variation of a load-related voltage or current as a function of the frequency of the excitation signal
IS S
IL 1 + 2 L
VS CD Player (Source) Amplifier (Circuit) A physical system Speakers (Load)
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