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b and c are spread out with perfect conductors as in Fig. 2.10. A loop is any closed path in a circuit.
A loop is a closed path formed by starting at a node, passing through a
set of nodes, and returning to the starting node without passing through
any node more than once. A loop is said to be independent if it contains a
branch which is not in any other loop. Independent loops or paths result
in independent sets of equations.
For example, the closed path abca containing the 2- resistor in
Fig. 2.11 is a loop. Another loop is the closed path bcb containing the
3- resistor and the current source. Although one can identify six loops
in Fig. 2.11, only three of them are independent.
A network with b branches, n nodes, and l independent loops will
satisfy the fundamental theorem of network topology:
b =l+n−1 (2.12) As the next two deﬁnitions show, circuit topology is of great value
to the study of voltages and currents in an electric circuit. Two or more elements are in series if they are cascaded or connected sequentially
and consequently carry the same current.
Two or more elements are in parallel if they are connected to the same two nodes
and consequently have the same voltage across them.
Elements are in series when they are chain-connected or connected sequentially, end to end. For example, two elements are in series if they
share one common node and no other element is connected to that common node. Elements in parallel are connected to the same pair of terminals. Elements may be connected in a way that they are neither in series
nor in parallel. In the circuit shown in Fig. 2.10, the voltage source and
the 5- resistor are in series because the same current will ﬂow through
them. The 2- resistor, the 3- resistor, and the current source are in
parallel because they are connected to the same two nodes (b and c)
and consequently have the same voltage across them. The 5- and 2resistors are neither in series nor in parallel with each other. EXAMPLE 2.4
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- Spring '12