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CHAPTER 4 Basic Network Laws and Theorems
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Problem 58 (a) (b) Going back to Fig. 59 in section 4.6, replace the network to the left of terminals a, b with the equivalent Norton generator. In Fig. 59, suppose RL 10 ohms. Show that the Thevenin and Norton generators produce the same external load current, IL 2:267 amperes, to three decimal places in each case.
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Problem 59 In Fig. 60, if Isc 6:155 amperes and Gg 0:109 mho, draw the Thevenin equivalent generator.
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The Method of Node Voltages
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The method of node voltages is a procedure for network analysis based upon Kirchho s current law (section 4.2). The procedure is as follows. We begin by selecting one of the nodes in the network to be the reference node. The unknown voltages, at the other nodes in the network, are to be found relative to the zero voltage at the reference node. Thus, in the node method of network analysis the NODE VOLTAGES ARE THE UNKNOWNS, instead of the currents as in the loop method. If a network has N nodes, then N 1 node voltages will be present (because one of the N nodes is selected to be the reference node, which is then taken to be at zero reference potential ). We begin our discussion with Fig. 61, which shows a resistance of R ohms connected between two node points a and b. Let us assume a current of I amperes owing through R from node a to node b, as shown.
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Fig. 61
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In Fig. 61, g is the reference node. It will always be understood, unless de nitely stated otherwise, that all node voltages in a network are given relative to the reference node, which is taken as being at zero voltage, that is, Vg 0. Thus, in Fig. 61, the node voltages Va and Vb are measured with respect to the reference node g. In practical work it is often necessary, or at least desirable, to connect one side of a circuit to earth or ground by connection, for example, to an underground water pipe. This may be done to insure the safety of personnel, or to minimize noise pickup, and so on. Often, however, we ll use the ground symbol as a convenient symbol to designate the reference node, even though it may not actually be connected to an earth ground. In Fig. 61 we are assuming that current is owing from left to right, from node a to node b, as shown. This indicates that node a is POSITIVE with respect to node b, because
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CHAPTER 4 Basic Network Laws and Theorems
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conventional current* ows from a point of higher potential to a point of lower potential, that is, from positive to negative. For example, if Va 100 volts and Vb 70 volts (both measured with respect to g), then node a would be 30 volts positive with respect to node b, so that current would ow from node a to node b, as in Fig. 61. Since, by Ohm s law, current equals voltage divided by resistance I V=R , the situation in Fig. 61 is stated algebraically by writing I Va Vb R 68
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In applying the method of node voltages to a network we rst designate the reference node, which we ll generally do by use of the ground symbol as mentioned above. We then label the unknown voltages, at the di erent nodes, as Va , Vb , Vc , and so on, all voltages being with respect to the zero voltage at the reference node. We next draw and label the current arrows I1 , I2 , I3 , and so on, at each node (see Fig. 44, section 4.2), and then write the current equation at each node (see eq. (56), section 4.2). Now, in each current equation, replace each current by its equivalent in terms of eq. (68); doing this gives us the required equations in terms of the unknown node voltages. If the battery voltages and resistance values are known, the resulting linear simultaneous equations can then be solved to nd the node voltages. Example
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In Fig. 62, resistance values are in ohms. Using the node-voltage procedure nd, to the third decimal place, Vb and Vc . Use the current arrows as given in the gure, or redraw them any way you wish.
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