3.2 Node and mesh analysis . David irwin In this section Kirchhoffs current law(KCL) and Kirchhoff's voltage law(KVL) will be used to determine currents and voltages throughout a network. For simplicity, we will first illustrate the basic principles of both node analysis and mesh analysis using only dc circuits. Once the fundamental concepts have been explained and illustrated, we will demonstrate the generality of both analysis techniques through an ac circuit exampl Node analysis In a node analysis, the node voltages are the variables in a circuit, v1=4VV2=4v and KCl is the vehicle used to determine them. One node in the network is selected as a reference node and then all other node voltages are defined with respect to that particular node. This refer ence node is typically referred to as ground using the symbol (4),20 indicating that it is at ground-zero potential Consider the network shown in Fig. 3.3. The network has three nodes, and the nodes at the bottom of the circuit has been selected as the reference node. Therefore the two remaining nodes, labeled VI and V2, are measured with respect to this reference node. Suppose that the node voltages V, and V, have somehow been FIGURE 3.3 A three-node network. determined, i. e, V=4 V and v2=-4 V. Once these node voltages known, Ohms law can be used to find all branch currents. For example, 2A V-V2_4-(-4) 4A 3 4A 1 Note that KCL is satisfied at every node, i. e, l2+8+I3=0 +6-8-1=0 Therefore, as a general rule, if the node voltages are known, all branch currents in the network can be immediately determined. In order to determine the node voltages in a network, we apply KCL to every node in the network except the reference node. There- V, fore, given an N-node circuit, we employ N-1 linearly independent 20 simultaneous equations to determine the n-1 unknown node volt 19 ages. Graph theory, which is covered in Section 3.6, can be used to 1 prove that exactly N-1 linearly independent KCL equations are required to find the N-1 unknown node voltages in a network. Let us now demonstrate the use of KCL in determining the node FIGURE 3.4 A four-node network. voltages in a network. For the network shown in Fig 3.4, the bottom c 2000 by CRC Press LLC© 2000 by CRC Press LLC 3.2 Node and Mesh Analysis J. David Irwin In this section Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL) will be used to determine currents and voltages throughout a network. For simplicity, we will first illustrate the basic principles of both node analysis and mesh analysis using only dc circuits. Once the fundamental concepts have been explained and illustrated, we will demonstrate the generality of both analysis techniques through an ac circuit example. Node Analysis In a node analysis, the node voltages are the variables in a circuit, and KCL is the vehicle used to determine them. One node in the network is selected as a reference node, and then all other node voltages are defined with respect to that particular node. This refer￾ence node is typically referred to as ground using the symbol ( ), indicating that it is at ground-zero potential. Consider the network shown in Fig. 3.3. The network has three nodes, and the nodes at the bottom of the circuit has been selected as the reference node. Therefore the two remaining nodes, labeled V1 and V2, are measured with respect to this reference node. Suppose that the node voltages V1 and V2 have somehow been determined, i.e., V1 = 4 V and v2 = –4 V. Once these node voltages are known, Ohm’s law can be used to find all branch currents. For example, Note that KCL is satisfied at every node, i.e., I1 – 6 + I2 = 0 –I2 + 8 + I3 = 0 –I1 + 6 – 8 – I3 = 0 Therefore, as a general rule, if the node voltages are known, all branch currents in the network can be immediately determined. In order to determine the node voltages in a network, we apply KCL to every node in the network except the reference node. There￾fore, given an N-node circuit, we employ N – 1 linearly independent simultaneous equations to determine the N – 1 unknown node volt￾ages. Graph theory, which is covered in Section 3.6, can be used to prove that exactly N – 1 linearly independent KCL equations are required to find the N – 1 unknown node voltages in a network. Let us now demonstrate the use of KCL in determining the node voltages in a network. For the network shown in Fig. 3.4, the bottom FIGURE 3.3 A three-node network. I V I V V I V 1 1 2 1 2 3 2 0 2 2 2 4 4 2 4 0 1 4 1 4 = - = = - = - - = = - = - = - A A A ( ) FIGURE 3.4 A four-node network