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Part I
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Circuits
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a' R3
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V1 + _
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RL V2 b
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R1 = 100
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Schematics, Diagrams, Circuits, and Given Data: V1 = 50 V; I = 05 A; V2 = 5 V;
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; R2 = 100
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; R3 = 200
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; R4 = 160
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Assumptions: Assume reference node is at the bottom of the circuit Analysis: First, we sketch the circuit again, to take advantage of the source
transformation technique; we emphasize the location of the nodes for this purpose, as shown in Figure 367 Nodes a and b have been purposely separated from nodes a and b even though these are the same pairs of nodes We can now replace the branch consisting of V1 and R1 , which appears between nodes a and b , with an equivalent Norton circuit with Norton current source V1 /R1 and equivalent resistance R1 Similarly, the series branch between nodes a and b is replaced by an equivalent Norton circuit with Norton current source V2 /R3 and equivalent resistance R3 The result of these manipulations is shown in Figure 368 The same circuit is now depicted in Figure 369 with numerical values substituted for each component Note how easy it is to visualize the equivalent resistance: if each current source is replaced by an open circuit, we nd: RT = R1 ||R2 ||R3 || + R4 = 200||100||100 + 160 = 200
a' R3
V1 + _
RL V2 b
V1 R1
V2 R3
3
Resistive Network Analysis
160
50 A 100
5 A 200
1 A 2
100
100
200
The calculation of the Norton current is similarly straightforward, since it simply involves summing the currents:
0025 A
200
iN = 05 0025 05 = 0025 A Figure 370 depicts the complete Norton equivalent circuit connected to the load
Comments: It is not always possible to reduce a circuit as easily as was shown in this
example by means of source transformations However, it may be advantageous to use source transformation as a means of converting parts of a circuit to a different form, perhaps more naturally suited to a particular solution method (eg, nodal analysis)
Experimental Determination of Thevenin and Norton Equivalents The idea of equivalent circuits as a means of representing complex and sometimes unknown networks is useful not only analytically, but in practical engineering applications as well It is very useful to have a measure, for example, of the equivalent internal resistance of an instrument, so as to have an idea of its power requirements and limitations Fortunately, Th venin and Norton equivalent circuits can also be e evaluated experimentally by means of very simple techniques The basic idea is that the Th venin voltage is an open-circuit voltage and the Norton current e is a short-circuit current It should therefore be possible to conduct appropriate measurements to determine these quantities Once vT and iN are known, we can determine the Th venin resistance of the circuit being analyzed according to the e relationship RT = vT iN (334)
How are vT and iN measured, then Figure 371 illustrates the measurement of the open-circuit voltage and shortcircuit current for an arbitrary network connected to any load and also illustrates that the procedure requires some special attention, because of the nonideal nature of any practical measuring instrument The gure clearly illustrates that in the presence of nite meter resistance, rm , one must take this quantity into account in the computation of the short-circuit current and open-circuit voltage; vOC and iSC appear between quotation marks in the gure speci cally to illustrate that the measured open-circuit voltage and short-circuit current are in fact affected by the internal resistance of the measuring instrument and are not the true quantities
Part I
Circuits
a Unknown network b An unknown network connected to a load a Load
A Unknown network i SC rm
b Network connected for measurement of short-circuit current a + Unknown network rm
vOC b
Network connected for measurement of open-circuit voltage
Figure 371 Measurement of open-circuit voltage and short-circuit current
You should verify that the following expressions for the true short-circuit current and open-circuit voltage apply (see the material on nonideal measuring instruments in Section 28): iN = iSC 1 + rm RT (335) RT vT = vOC 1 + rm where iN is the ideal Norton current, vT the Th venin voltage, and RT the true e Th venin resistance If you recall the earlier discussion of the properties of ideal e ammeters and voltmeters, you will recall that for an ideal ammeter, rm should approach zero, while in an ideal voltmeter, the internal resistance should approach an open circuit (in nity); thus, the two expressions just given permit the determination of the true Th venin and Norton equivalent sources from an (imperfect) e measurement of the open-circuit voltage and short-circuit current, provided that the internal meter resistance, rm , is known Note also that, in practice, the internal resistance of voltmeters is suf ciently high to be considered in nite relative to the equivalent resistance of most practical circuits; on the other hand, it is impossible to construct an ammeter that has zero internal resistance If the internal ammeter resistance is known, however, a reasonably accurate measurement of short-circuit current may be obtained The following example illustrates the point
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