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CIRCUIT CONCEPTS
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The circuit element that stores energy in a magnetic eld is an inductor (also called an inductance). With time-variable current, the energy is generally stored during some parts of the cycle and then returned to the source during others. When the inductance is removed from the source, the magnetic eld will collapse; in other words, no energy is stored without a connected source. Coils found in electric motors, transformers, and similar devices can be expected to have inductances in their circuit models. Even a set of parallel conductors exhibits inductance that must be considered at most frequencies. The power and energy relationships are as follows. ! di d 1 2 i Li p vi L dt dt 2 t2 t2 1 2 2 wL p dt Li dt L i2 i1 2 t1 t1 Energy stored in the magnetic eld of an inductance is wL 1 Li2 . 2
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EXAMPLE 2.2. In the interval 0 > t > =50 s a 30-mH inductance has a current i 10:0 sin 50t (A). Obtain the voltage, power, and energy for the inductance. v L di 15:0 cos 50t V dt t p vi 75:0 sin 100t W wL
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p dt 0:75 1 cos 100t J
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As shown in Fig. 2-7, the energy is zero at t 0 and t =50 s. Thus, while energy transfer did occur over the interval, this energy was rst stored and later returned to the source.
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CIRCUIT CONCEPTS
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The circuit element that stores energy in an electric eld is a capacitor (also called capacitance). When the voltage is variable over a cycle, energy will be stored during one part of the cycle and returned in the next. While an inductance cannot retain energy after removal of the source because the magnetic eld collapses, the capacitor retains the charge and the electric eld can remain after the source is removed. This charged condition can remain until a discharge path is provided, at which time the energy is released. The charge, q Cv, on a capacitor results in an electric eld in the dielectric which is the mechanism of the energy storage. In the simple parallel-plate capacitor there is an excess of charge on one plate and a de ciency on the other. It is the equalization of these charges that takes place when the capacitor is discharged. The power and energy relationships for the capacitance are as follows. ! dv d 1 p vi Cv Cv2 dt dt 2 t2 t2 1 wC p dt Cv dv C v2 v2 2 1 2 t1 t1 The energy stored in the electric eld of capacitance is wC 1 Cv2 . 2
EXAMPLE 2.3. In the interval 0 > t > 5 ms, a 20-mF capacitance has a voltage v 50:0 sin 200t (V). Obtain the charge, power, and energy. Plot wC assuming w 0 at t 0. q Cv 1000 sin 200t mC dv 0:20 cos 200t A dt p vi 5:0 sin 400t W t2 p dt 12:5 1 cos 400t mJ wC i C
In the interval 0 > t > 2:5 ms the voltage and charge increase from zero to 50.0 V and 1000 mC, respectively. Figure 2-8 shows that the stored energy increases to a value of 25 mJ, after which it returns to zero as the energy is returned to the source.
Fig. 2-8
CIRCUIT DIAGRAMS
Every circuit diagram can be constructed in a variety of ways which may look di erent but are in fact identical. The diagram presented in a problem may not suggest the best of several methods of solution. Consequently, a diagram should be examined before a solution is started and redrawn if necessary to show more clearly how the elements are interconnected. An extreme example is illustrated in Fig. 2-9, where the three circuits are actually identical. In Fig. 2-9(a) the three junctions labeled A
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