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Norton’s Theorem

This theorem is in fat, an alternative to the thevenin’s theorem. Where as by Thevenin’s theorem a complex two-terminal network may be simplified for solution by reducing it into a simple circuit in which the so called open circuit voltage and the looking back resistance are connected in series with the load resistor. By Norton’s theorem network is reduced into a simple circuit in which a parallel combination of constant current source and looking-back resistance feeds the load resistance.

In both theorem use of the resistance looking back into the network form the load resistor terminals, with all sources of e.m.f. removed leaving their internal resistances in the circuit, is made. However while solving circuit by Thevnin’s theorem, the open circuit voltage is determined at the load terminal with the load removed whereas in Norton’s method use of a fictitious constant current source is made, the constant current delivered being equal to the current hat would pass into a short-circuit connected across the output terminals of the given network.

Now for understanding this theorem let us considers a circuit shown in which load current I1. is to be determined.

The current in any passive circuit element (which may be called R1) in a network is the same as would flow in it if it were connected in parallel with R and the parallel pair were supplied with a constant current Isc, R is the resistance measured “looking back” into the original circuit after R1 has be disconnected and all the sources have been replaced by their internal resistance Isc s the current which will flow in a short-circuit at the terminals of Rn in the orginal circuit.

 If short-circuit is placed between terminals A and B then short circuit current will be given by the expression

Isc =         E                R2      =               ER2           
       (r+R1)+ R2R3       R2+R3        (r+R1)(R2+R3)+R2R3

The resistance of the network measured between terminals A and B,

R = R3 +       1        = R3 + (R1+r)R2  = R2R3+(R1+r)(R2+R3)
              1    +  1              R3+R1+r            R2+R1+r
            (R1+r)  R2

and load current through load resistance, I1is give by the expression

                              ER2                 X  R2R3+(R1+ r)(R2+R3)    
IL = Isc R (r+R1)(R2+R3)+R2R3              R2+R1+ r         
                                R2R3+(R1+ r)(R2+R3) + RL
                                      R2+R1+ r
              =                      ER2                            
                  R2R3+(R1+ r)(R2+R3)+ RL (R2+R1+ r)

Norton’s equivalent circuit is shown in.

The Norton’s theorem may be stated as follows:

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