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Fig. 62
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Solution First, for the node at Vb : I1 I2 I3 0 and, for the node at Vc : I3 I4 I5 0 B A
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* Conventional current is the ow of positive charge (section 2.1), which creates a to voltage drop across a resistance in the direction of the current ow.
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CHAPTER 4 Basic Network Laws and Theorems
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Next, by inspection we see that Va 20 volts and Vd 12 volts; thus, applying eq. (68), we have I1 20 Vb 6 I3 Vb Vc 2 I5 12 Vc 3
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I2 Vb =4
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I4 Vc =5
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Now, upon substituting these values into equations (A) and (B), you should nd that 11Vb 6Vc 40 15Vb 31Vc 120 and thus the answers are Vb 7:809 V and Vc 7:649 V
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Problem 60 In Fig. 63, use the node-voltage procedure to nd the voltages at nodes 1 and 2. Resistance values are in ohms.
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Fig. 63
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Sinusoidal Waves. rms Value. As Vector Quantities
5.1 Introduction
So far in our work we ve dealt only with currents and voltages whose senses of direction and polarity never change. Such currents and voltages are called DIRECT currents and DIRECT voltages. Thus, since the polarity of a battery does not change, we say that a battery is a source of direct voltage, producing a ow of direct current. The abbreviation dc is commonly used to identify such quantities; thus we have dc voltage, dc current, dc power, and so on. There is, however, another class of voltages and currents whose senses of both direction and polarity continually ALTERNATE with time, plus to minus, minus to plus, and so on, endlessly. The term alternating is used to denote such a voltage or current, and the abbreviation ac is used to denote such quantities. Thus we have ac voltage, ac current, ac power, and so on. O hand, a person might expect that the algebra of ac circuits would be considerably di erent, and more di cult, than the algebra of dc circuits. It is, of course, true that ac calculations can di er greatly from dc calculations. But such di erences can be concisely expressed mathematically, and, when this is done, we ll nd that the form of the algebraic statements (Kirchho s laws, loop currents, and so on) that we learned in dc work will carry over directly into our ac work. Let us now begin the study of this most interesting and useful subject.
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CHAPTER 5 Sinusoidal Waves. rms Value
The Sinusoidal Functions and the Tangent Function
Fundamental to the study of alternating currents are the SINUSOIDAL functions, the term sinusoidal ( sign u SOID al ) denoting either the SINE function or the COSINE function. The sine and cosine are simple but remarkable functions, having properties unlike any others in the entire realm of mathematics. We ll nd that the two functions (sine and cosine) can be said to have identical waveforms, the only di erence being that the two are shifted with respect to each other on the horizontal or time axis. As a matter of fact, a person viewing a sinusoidal waveform on an oscilloscope, and having no other information, could not say whether the waveform represented a sine or a cosine function. Let us now proceed with some de nitions. We rst meet the sine and cosine in the study of plane trigonometry, in connection with the geometry of a RIGHT TRIANGLE, using the terminology of Fig. 64.
Fig. 64
In the triangle, the side h, opposite the 908 angle, is called the hypotenuse, as usual in a right triangle. We ll call the angle  (theta) the reference angle ; then the side opposite  is called the OPPOSITE side and the side adjacent to  is called the ADJACENT side, as shown. Remember that we always have a RIGHT triangle, so that one angle always remains xed at 908. Note that we ve denoted the third angle by  (phi or fee ). Since the three angles of a plane triangle must add up to 180 degrees, it follows that  90  degrees. An important fact concerning Fig. 64 (or any triangle, for that matter) can be understood as follows. Imagine that we looked at Fig. 64 through a magnifying glass having a magnifying power of k times. Doing this would change the apparent SIZE of the triangle (each of the three sides would be multiplied by k), but it would not change the SHAPE of the triangle in any way; that is, it would not change the ANGLES in any way. This will serve to illustrate the important fact that In Fig. 64 the RATIO of any one side to either of the other two sides depends not upon the SIZE of the triangle but only upon the ANGLE . (The angle  is automatically known if the reference angle  is given, because  90 .) In the study of alternating currents we deal mainly with three di erent ratios of the sides of a right triangle. In terms of an angle , the three ratios are called the SINE ( sign ) of
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