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= 450 0 V Vb = 440 V 6 = 420 349 V = 7795 0098 V Vcd = 1539 12 V = 2306 1875 V
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If the ground is now moved from node e to node d, determine Vb and Vbc
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a Z4 + + V _ _ S1 e + VS2 _ Z5 d c Z1 Z3 b
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463 The mesh currents and node voltages in the circuit
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shown in Figure P463 are: i1 (t) = 3127 i2 (t) = 3914 i3 (t) = 1900 v1 (t) = 1300 v2 (t) = 1300 cos( t cos( t cos( t cos( t cos( t 0825) A 178) A + 0655) A + 0176) V 0436) V
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where = 3770 rad/s Determine one of the following L1 , C2 , R3 , or L3
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1 Z4 + + V _ _S1 L1 I1 I3 + + V _ S2 _ Z5 C2 I2 2 L3 R3
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Figure P461
462 Determine Vo in the circuit of Figure P462 if:
vi = 4 cos(1,000t + ) V 6 L = 60 mH C = 125 F RL = 120
L + + _ Vi C C RL + V o
Figure P463
Figure P462
Transient Analysis
51 INTRODUCTION
The aim of this chapter is to explore the solution of circuits that contain resistances, inductances, capacitances, voltage and current sources, and switches The response of a circuit to the sudden application of a voltage or current is called transient response The most common instance of a transient response in a circuit occurs when a switch is turned on or off a rather common event in electrical circuits Although there are many possible types of transients that can be introduced in a circuit, in the present chapter we shall focus exclusively on the transient response of circuits in which a switch activates or deactivates a DC source Further, we shall restrict our analysis, for the sake of simplicity, to rst- and second-order transients, that is to circuits that have only one or two energy storage elements The graphs of Figure 51 illustrate the result of the sudden appearance of a voltage across a hypothetical load [a DC voltage in Figure 51(a), an AC voltage in Figure 51(b)] In the gure, the source voltage is turned on at time t = 02 s The voltage waveforms of Figure 51 can be subdivided into three regions: a steady-state region, for 0 t 02 s; a transient region for 02 t 2 s (approximately); and a new steady-state region for t > 2 s, where the voltage reaches a steady DC or AC condition The objective of transient analysis is to describe the behavior of a voltage or a current during the transition that takes place between two distinct steady-state conditions
5
Transient Analysis
1 08 Volts 06 04 02 0 0 02 04 06 10 12 14 t (s) (a) Transient DC voltage 08 16 18 20
1 05 Volts 0
05 1
10 12 14 t (s) (b) Transient sinusoidal voltage
Figure 51 Examples of transient response
t=0 Switch 12 V
Complex load
Figure 52 Circuit with switched DC excitation
Switch t=0 Circuit containing RL/RC combinations
Figure 53 A general model of the transient analysis problem
You already know how to analyze circuits in a sinusoidal steady state by means of phasors The material presented in the remainder of this chapter will provide the tools necessary to describe the transient response of circuits containing resistors, inductors, and capacitors A general example of the type of circuit that will be discussed in this section is shown in Figure 52 The switch indicates that we turn the battery power on at time t = 0 Transient behavior may be expected whenever a source of electrical energy is switched on or off, whether it be AC or DC A typical example of the transient response to a switched DC voltage would be what occurs when the ignition circuits in an automobile are turned on, so that a 12-V battery is suddenly connected to a large number of electrical circuits The degree of complexity in transient analysis depends on the number of energy-storage elements in the circuit; the analysis can became quite involved for high-order circuits In this chapter, we shall analyze only rstand second-order circuits that is, circuits containing one or two energy-storage elements, respectively In electrical engineering practice, we would typically resort to computer-aided analysis for higher-order circuits A convenient starting point in approaching the transient response of electrical circuits is to consider the general model shown in Figure 53, where the circuits in the box consist of a combination of resistors connected to a single energy-storage element, either an inductor or a capacitor Regardless of how many resistors the circuit contains, it is a rst-order circuit In general, the response of a rstorder circuit to a switched DC source will appear in one of the two forms shown in Figure 54, which represent, in order, a decaying exponential and a rising exponential waveform In the next sections, we will systematically analyze these responses by recognizing that they are exponential in nature and can be computed very easily once we have the proper form of the differential equation describing the circuit
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