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Print QR Code JIS X 0510 in Software Copyright 2003, 1997, 1986, 1965 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

Copyright 2003, 1997, 1986, 1965 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
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CIRCUIT CONCEPTS
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[CHAP. 2
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In general, a coil can be represented by either a series or a parallel arrangement of circuit elements. The frequency of the applied voltage may require that one or the other be used to represent the device.
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A voltage function and a polarity must be speci ed to completely describe a voltage source. The polarity marks, and , are placed near the conductors of the symbol that identi es the voltage source. If, for example, v 10:0 sin !t in Fig. 2-4(a), terminal A is positive with respect to B for 0 > !t > , and B is positive with respect to A for  > !t > 2 for the rst cycle of the sine function.
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Similarly, a current source requires that a direction be indicated, as well as the function, as shown in Fig. 2-4(b). For passive circuit elements R, L, and C, shown in Fig. 2-4(c), the terminal where the current enters is generally treated as positive with respect to the terminal where the current leaves. The sign on power is illustrated by the dc circuit of Fig. 2-5(a) with constant voltage sources VA 20:0 V and VB 5:0 V and a single 5- resistor. The resulting current of 3.0 A is in the clockwise direction. Considering now Fig. 2-5(b), power is absorbed by an element when the current enters the element at the positive terminal. Power, computed by VI or I 2 R, is therefore absorbed by both the resistor and the VB source, 45.0 W and 15 W respectively. Since the current enters VA at the negative terminal, this element is the power source for the circuit. P VI 60:0 W con rms that the power absorbed by the resistor and the source VB is provided by the source VA .
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CHAP. 2]
CIRCUIT CONCEPTS
Fig. 2-5
VOLTAGE-CURRENT RELATIONS
The passive circuit elements resistance R, inductance L, and capacitance C are de ned by the manner in which the voltage and current are related for the individual element. For example, if the voltage v and current i for a single element are related by a constant, then the element is a resistance, R is the constant of proportionality, and v Ri. Similarly, if the voltage is the time derivative of the current, then the element is an inductance, L is the constant of proportionality, and v L di=dt. Finally, if the current in the element is the time derivative of the voltage, then the element is a capacitance, C is the constant of proportionality, and i C dv=dt. Table 2-1 summarizes these relationships for the three passive circuit elements. Note the current directions and the corresponding polarity of the voltages.
Table 2-1 Circuit element Units Voltage Current Power
ohms ()
v Ri (Ohms s law)
p vi i2 R
Resistance
henries (H)
di dt
v dt k1 p vi Li
di dt
Inductance
farads (F)
i dt k2 i C
dv dt
p vi Cv
dv dt
Capacitance
CIRCUIT CONCEPTS
[CHAP. 2
RESISTANCE
All electrical devices that consume energy must have a resistor (also called a resistance) in their circuit model. Inductors and capacitors may store energy but over time return that energy to the source or to another circuit element. Power in the resistor, given by p vi i2 R v2 =R, is always positive as illustrated in Example 2.1 below. Energy is then determined as the integral of the instantaneous power t2 t2 1 t2 2 2 w p dt R i dt v dt R t1 t1 t1
EXAMPLE 2.1. A 4.0- resistor has a current i 2:5 sin !t (A). Find the voltage, power, and energy over one cycle. ! 500 rad/s. v Ri 10:0 sin !t V p vi i2 R 25:0 sin2 !t W ! t t sin 2!t J w p dt 25:0 2 4! 0 The plots of i, p, and w shown in Fig. 2-6 illustrate that p is always positive and that the energy w, although a function of time, is always increasing. This is the energy absorbed by the resistor.
Fig. 2-6
CHAP. 2]
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