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A linear network which contains two or more independent sources can be analyzed to obtain the various voltages and branch currents by allowing the sources to act one at a time, then superposing the results. This principle applies because of the linear relationship between current and voltage. With dependent sources, superposition can be used only when the control functions are external to the network containing the sources, so that the controls are unchanged as the sources act one at a time. Voltage sources to be suppressed while a single source acts are replaced by short circuits; current sources are replaced by open circuits. Superposition cannot be directly applied to the computation of power, because power in an element is proportional to the square of the current or the square of the voltage, which is nonlinear. As a further illustration of superposition consider equation (7) of Example 4.4:       11 21 31 V2 V3 I1 V 1 R R R which contains the superposition principle implicitly. Note that the three terms on the right are added to result in current I1 . If there are sources in each of the three meshes, then each term contributes to the current I1 . Additionally, if only mesh 3 contains a source, V1 and V2 will be zero and I1 is fully determined by the third term.
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EXAMPLE 4.7 Compute the current in the 23- resistor of Fig. 4-11(a) by applying the superposition principle. With the 200-V source acting alone, the 20-A current source is replaced by an open circuit, Fig. 4-11(b).
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When the 20-A source acts alone, the 200-V source is replaced by a short circuit, Fig. 4-11(c). resistance to the left of the source is 27 47 21:15  Req 4 74   21:15 00 20 9:58 A I23 21:15 23
Then The total current in the 23- resistor is
0 00 I23 I23 I23 11:23 A
THEVENIN S AND NORTON S THEOREMS
A linear, active, resistive network which contains one or more voltage or current sources can be replaced by a single voltage source and a series resistance (The venin s theorem), or by a single current source and a parallel resistance (Norton s theorem). The voltage is called the The venin equivalent voltage, V 0 , and the current the Norton equivalent current, I 0 . The two resistances are the same, R 0 . When terminals ab in Fig. 4-12(a) are open-circuited, a voltage will appear between them.
Fig. 4-12
From Fig. 4-12(b) it is evident that this must be the voltage V 0 of the Thevenin equivalent circuit. If a short circuit is applied to the terminals, as suggested by the dashed line in Fig. 4-12(a), a current will result. From Fig. 4-12(c) it is evident that this current must be I 0 of the Norton equivalent circuit. Now, if the circuits in (b) and (c) are equivalents of the same active network, they are equivalent to each other. It follows that I 0 V 0 =R 0 . If both V 0 and I 0 have been determined from the active network, then R 0 V 0 =I 0 .
EXAMPLE 4.8 Obtain the Thevenin and Norton equivalent circuits for the active network in Fig. 4-13(a). With terminals ab open, the two sources drive a clockwise current through the 3- and 6- resistors [Fig. 4-13(b)]. I 20 10 30 A 3 6 9
Since no current passes through the upper right 3- resistor, the Thevenin voltage can be taken from either active branch:
ANALYSIS METHODS
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Fig. 4-13 Vab V 0 20 or Vab   30 3 10 V 9   30 6 10 10 V V0 9
The resistance R 0 can be obtained by shorting out the voltage sources [Fig. 4.13(c)] and nding the equivalent resistance of this network at terminals ab: R0 3 3 6 5 9 Assuming that it
When a short circuit is applied to the terminals, current Is:c: results from the two sources. runs through the short from a to b, we have, by superposition, 2 3 2 3     6 20 3 10 6 6 7 7 Is:c: I 0 4 5 4 5 2A 3 6 3 3 6 3 3 3 3 6 9 6
Figure 4-14 shows the two equivalent circuits. In the present case, V 0 , R 0 , and I 0 were obtained independently. Since they are related by Ohm s law, any two may be used to obtain the third.
Fig. 4-14
The usefulness of Thevenin and Norton equivalent circuits is clear when an active network is to be examined under a number of load conditions, each represented by a resistor. This is suggested in
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