# 13 n1 where n is the number of branches connected to

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online