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CHAPTER 3 Determinants and Equations
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IF and only if eq. (54) is satis ed, can the system of eq. (53) have solutions other than the trivial x y z 0. Now compare eq. (54) with the equation ax by cz 0 from eq. (53); the comparison shows that if eq. (54) is satis ed, then eq. (53) is satis ed for the non-trivial values       d e  d f  e f   ; ; 55 z  y  x  g h  g i  h i  The procedure can be extended to homogeneous linear systems of any order. Problem 44 Given the homogeneous linear system 3x 2y 5z 0 x y z 0 2x y 4z 0 Does the system possess non-trivial solutions If so, nd such a solution. Problem 45 Given the homogeneous linear system 4x 18y 7z 0 2x 4py pz 0 px 3y 5z 0 where p is constant, nd the values of p for which the system has non-trivial solutions, and nd such a solution.
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Basic Network Laws and Theorems
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4.1 Introduction
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In this chapter we continue with the work we began in Chap. 2. While the fundamental procedures of Chap. 2 are very useful they can, in some cases, become quite awkward to use. Also, there are some types of networks that cannot be separated into purely series and parallel groups of resistors, and in such cases the procedures cannot be used. The so-called bridge networks are of this type. It is therefore necessary that we have available a more general method of circuit analysis than that used in Chap. 2. Fortunately such a procedure exists, resting upon what are generally called Kirchho s current and voltage laws (usually pronounced as KIRK o ), which are the subject of this chapter.
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Kirchhoff s Current Law
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Kirchho s current law is based upon the fact that at any connecting point in a network the sum of the currents owing toward the point is equal to the sum of the currents owing away from the point. The law is illustrated in the examples in Figs. 42 and 43, where the arrows show the directions in which it is given that the currents are owing. (The number alongside each arrow is the amount of current associated with that arrow.) The example of Fig. 42 shows that if a current of 8 amperes is owing toward a connecting point p, then it is true that 8 amperes has to be owing away from p. In the same way, Fig. 43 shows that if 11 amperes is owing toward connection point p, then 11 amperes has to be owing away from p.
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CHAPTER 4 Basic Network Laws and Theorems
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Fig. 42
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Fig. 43
Any connecting point, such as p in the above gures, is called a node* (as in load ), and the relationship at a node, or nodal point, is summarized in Kirchho s current law: The sum of the currents owing TO a node point equals the sum of the currents owing FROM that point. The currents owing into and out of a node point are called branch currents. Thus, in Fig. 42 the branch currents are 6, 2, and 8 amperes. There is an important point to be made in regard to branch currents, which is explained with the aid of Figs. 44 and 45, in which the branch currents are denoted by I1 , I2 , and I3 , as follows.
Fig. 44
Fig. 45
In Fig. 44 note that separate notation is used to denote the value of each of the three branch currents. However, by Kirchho s current law, I3 I1 I2 , and thus, as shown in Fig. 45, we need to use only two current designations. In other words, if we know any two of the three currents in Fig. 44, we can then nd the third current. In the same way, if there are, say, four branch currents entering and leaving a node point, and if we know any three of the currents, we can then nd the fourth current, and so on. It is important to note that Kirchho s current law can also be stated in terms of the algebraic sum of the currents at a junction or nodal point. This can be understood by referring to Fig. 44, as follows. In Fig: 44: thus: I1 I2 I3 I1 I2 I3 0 56
Equation 56 can also be arrived at by requiring that all current values at a junction point be put on one side of the current equation, and then requiring that currents owing TO the point be listed as positive currents and currents owing AWAY from the point be listed as negative currents. If this rule is understood to always apply, then Kirchho s current law can be stated in the form The algebraic sum of the currents at a node (junction point) is equal to zero.