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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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- Spring '12