FIGURE 3.1 Graph representation of a linear circuit. connection between two or more physical components; an edge is drawn as a line and represents a path,or branch, for current flow through a component(see Fig 3.1) The edges, or branches, of the graph are assigned current labels, i,, iz,..., im. Each current has a designated direction, usually denoted by an arrow symbol. If the arrow is drawn toward a node, the associated current is said to be entering the node; if the arrow is drawn away from the node, the current is said to be leaving the node. The current i is entering node b in Fig 3. 1; the current is is leaving node e. Given a branch, the pair of nodes to which the branch is attached defines the convention for measuring voltages in the circuit. Given the ordered pair of nodes(a, b), a voltage measurement is formed as follows: where va and v, are the absolute electrical potentials(voltages)at the respective nodes, taken relative to some reference node. Typically, one node of the circuit is labeled as ground, or reference node; the remaining nodes re assigned voltage labels. The measured quantity, vab, is called the voltage drop from node a to node b. we is called the voltage rise from a to b. Each node voltage implicitly defines the voltage drop between the respective node and the ground node. The pair of nodes to which an edge is attached may be written as(a, b)or(b, a). Given an ordered pair of nodes(a, b), a path from a to b is a directed sequence of edges in which the first edge in the sequence contains node label a, the last edge in the sequence contains node label b, and the node indices of any two adjacent members of the sequence have at least one node label in common. In Fig. 3.1, the edge sequence (ej, e2, e is not a path, because e, and e do not share a common node label. The sequence le, e,) is a path from node a to node c a path is said to be closed if the first node index of its first edge is identical to the second node index of its last edge. The following edge sequence forms a closed path in the graph given in Fig. 3.1: leu, ex, e,, ea, e,, Note that the edge sequences fes) and fen, el are closed paths Kirchhoff's Current law Kirchhoff's current law(KCL) imposes constraints on the currents in the branches that are attached to each node of a circuit. In simplest terms, KCL states that the sum of the currents that are entering a given node c 2000 by CRC Press LLC© 2000 by CRC Press LLC connection between two or more physical components; an edge is drawn as a line and represents a path, or branch, for current flow through a component (see Fig. 3.1). The edges, or branches, of the graph are assigned current labels, i1, i2, . . ., im. Each current has a designated direction, usually denoted by an arrow symbol. If the arrow is drawn toward a node, the associated current is said to be entering the node; if the arrow is drawn away from the node, the current is said to be leaving the node. The current i1 is entering node b in Fig. 3.1; the current i5 is leaving node e. Given a branch, the pair of nodes to which the branch is attached defines the convention for measuring voltages in the circuit. Given the ordered pair of nodes (a, b), a voltage measurement is formed as follows: vab = va – vb where va and vb are the absolute electrical potentials (voltages) at the respective nodes, taken relative to some reference node. Typically, one node of the circuit is labeled as ground, or reference node; the remaining nodes are assigned voltage labels. The measured quantity, vab, is called the voltage drop from node a to node b. We note that vab = –vba and that vba = vb – va is called the voltage rise from a to b. Each node voltage implicitly defines the voltage drop between the respective node and the ground node. The pair of nodes to which an edge is attached may be written as (a,b) or (b,a). Given an ordered pair of nodes (a, b), a path from a to b is a directed sequence of edges in which the first edge in the sequence contains node label a, the last edge in the sequence contains node label b, and the node indices of any two adjacent members of the sequence have at least one node label in common. In Fig. 3.1, the edge sequence {e1, e2, e4} is not a path, because e2 and e4 do not share a common node label. The sequence {e1, e2} is a path from node a to node c. A path is said to be closed if the first node index of its first edge is identical to the second node index of its last edge. The following edge sequence forms a closed path in the graph given in Fig. 3.1: {e1, e2, e3, e4, e6, e7}. Note that the edge sequences {e8} and {e1, e1} are closed paths. Kirchhoff’s Current Law Kirchhoff’s current law (KCL) imposes constraints on the currents in the branches that are attached to each node of a circuit. In simplest terms, KCL states that the sum of the currents that are entering a given node FIGURE 3.1 Graph representation of a linear circuit