13 n1 where n is the number of branches connected to

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Unformatted text preview: f currents entering a node (or a closed boundary) is zero. Mathematically, KCL implies that N in = 0 (2.13) n=1 where N is the number of branches connected to the node and in is the nth current entering (or leaving) the node. By this law, currents entering a node may be regarded as positive, while currents leaving the node may be taken as negative or vice versa. To prove KCL, assume a set of currents ik (t), k = 1, 2, . . . , flow into a node. The algebraic sum of currents at the node is iT (t) = i1 (t) + i2 (t) + i3 (t) + · · · i5 i1 Integrating both sides of Eq. (2.14) gives qT (t) = q1 (t) + q2 (t) + q3 (t) + · · · i4 i2 (2.14) (2.15) where qk (t) = ik (t) dt and qT (t) = iT (t) dt . But the law of conservation of electric charge requires that the algebraic sum of electric charges at the node must not change; that is, the node stores no net charge. Thus qT (t) = 0 → iT (t) = 0, confirming the validity of KCL. Consider the node in Fig. 2.16. Applying KCL gives i3 Figure 2.16 Currents at a node illustrating KCL. i1 + (−i2 ) + i3 + i4 + (−i5 ) = 0 (2.16) since currents i1 , i3 , and i4 are entering the node, while currents i2 and i5 are leaving it. By rearranging the terms, we get Closed boundary i1 + i3 + i4 = i2 + i5 (2.17) Equation (2.17) is an alternative form of KCL: The sum of the currents entering a node is equal to the sum of the currents leaving the node. Figure 2.17 Applying KCL to a closed boundary. | v v Two sources (or circuits in general) are said to be equivalent if they have the same i-v relationship at a pair of terminals. | e-Text Main Menu Note that KCL also applies to a closed boundary. This may be regarded as a generalized case, because a node may be regarded as a closed surface shrunk to a point. In two dimensions, a closed boundary is the same as a closed path. As typically illustrated in the circuit of Fig. 2.17, the total current entering the closed surface is equal to the total current leaving the surface. A simple application of KCL is combining current sources in...
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