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CHAPTER 4 Basic Network Laws and Theorems
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Notice that our generator delivers an almost constant current of 10 amperes, regardless of whether it works into a load of zero ohms or one million ohms. We thus see that a true constant-current generator is a theoretical device having in nitely great generated voltage but in nitely great internal resistance, the ratio of the two being equal to a nite constant current. The symbol for a constant-current generator is shown below, where I is the value of the constant current, the arrow designating the direction of positive current.
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Let us now return to the two-terminal network inside the box of Fig. 58, to which a load resistance RL can be connected, and take up the details of Norton s theorem. Norton s theorem is expressed in terms of the short-circuit current delivered by the network, and in terms of conductances instead of resistances. This makes Norton s theorem especially useful in the study of parallel circuits. The statement of Norton s theorem is as follows, after which we ll give the proof of the statement. The current in any load conductance GL , when connected to two terminals of a network, is the same as if GL were connected to a constant-current generator whose constant current is equal to the current that ows between the two terminals when they are short-circuited together, this constant-current generator then being put in parallel with a conductance equal to the conductance seen looking back into the open-circuited terminals of the network. (In this last step, all generators are removed and replaced with conductances equal to their internal conductances.) Norton s theorem is summarized graphically in Fig. 60, where Isc is the short-circuit current that ows from the network when terminals a, b are shorted together. Gg is the conductance seen looking back into the network with the terminals open-circuited, that is, with the switch open. Gg is the reciprocal of Rg in Thevenin s theorem.
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Fig. 60
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The truth of Norton s theorem can be shown as follows. Let any two-terminal network be inside the box of Fig. 58. We know that, as far as the external load resistance RL is concerned, the network inside the box can be replaced with the Thevenin equivalent
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CHAPTER 4 Basic Network Laws and Theorems
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generator of Fig. 57, where, by inspection of Fig. 57, we have IL Vg Rg RL 65
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Now put a short-circuit (a copper wire) between terminals a, b in Fig. 57. The shortcircuit current owing between the terminals is then (since RL 0 for this condition) Isc Vg Rg
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Now put this value of Vg , Vg Isc Rg , into eq. (65), then multiply both sides by RL . Since RL IL the voltage across the load VL , we get   Rg RL VL Isc Rg RL Now multiply the numerator and denominator of the last fraction by 1=Rg RL . Then, by the de nition of conductance (eq. (58)), the last equation becomes VL Isc Gg GL 66
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We now know that eq. (66) is the correct equation for the voltage VL appearing across the load. With this in mind, turn now to the proposed equivalent circuit of Fig. 60. Remembering that conductances in parallel add together like resistances in series (eq. (61)), and also remembering the basic relation, I GV (eq. (59)), we have, for Fig. 60, Isc Gg GL VL , so that VL Isc Gg GL
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Since the last equation is the same as eq. (66), it follows that Figs. 57 and 60 produce completely equal results as far as any external load is concerned, and therefore either can be used. TO SUMMARIZE: Any two-terminal network consisting of generators and linear* bilateral* resistances can be replaced as far as an external load connected to the two terminals is concerned by either a Thevenin generator (Fig. 57) or a Norton generator (Fig. 60). If the external load consists of multiple parallel branches, it will generally be more convenient to use the Norton generator. Problem 57 In Fig. 60, let IL denote the value of the current that would ow in the load conductance GL if the switch were closed. Now, by making use of the basic relationship I GV (eq. (59)), show that IL GL Isc Gg GL 67
* Recall that a resistance is linear if its value is independent of the amount of voltage applied to it or the amount of current owing in it. It is bilateral if current is able to ow through it equally well in both directions.