REAL GENERATORS 



Remembering that the relationship which equates the two arrangements is 

 E = Ir, if Ir be substituted into the first column wherever E appears, the two 

 columns will be found to be identical. The two representations are therefore 

 equivalent. 



Having established that a real generator may be represented in either of 

 two ways, it remains to decide which way is better in a particular case. 

 Consider a real generator of internal resistance r, and suppose it has a 

 measured open-circuit voltage E. Let us plot the output voltage and current 

 as a load resistance is made less than, equal to, and greater than r. The 

 relevant equations are, as we have seen, Kout = E{Rl{r + R)] and /out = 

 £/(/• + R) {Graph I). 



Clearly when R^ r, the output voltage is rather constant but the current 

 is not. And when R<^r, the output current is rather constant but the 

 voltage is not. Therefore, 



If a real generator is feeding a load of resistance much greater than its 

 own, the ideal constant-voltage generator symbol is appropriate, with the 

 internal resistance in series. If a real generator is feeding a load of resis- 

 tance much lower than its own, the ideal constant-voltage generator symbol 

 is appropriate, with the internal resistance in shunt. 



If the load resistance and generator resistance are of the same order, 

 either representation will do, but the constant-voltage symbol seems to be 

 commoner. A real generator having r<^R will be said to be of the constant- 

 voltage type. A real generator having r^ R will be said to be of the constant- 

 current type. 



Maximum power transfer theorem 



In Figure 2.6 we have a real generator feeding a load. An important 

 problem is to decide what value of R extracts maximum power from the 

 generator terminals. 



The load current is Ejir + R), so the load power, P, is PR = {E'^R]l{{r + 

 Rf). When the load power is a maximum, the rate of change of load power 

 with load resistance will be zero, i.e. 



d^ ^ 



but 'a^eA^p:^.^ ' 



dR \{r-^Rf^{r + Rf] 



whence R = r. ^'^'"-^ ^-^ 



A load is said to be matched to a generator, and takes maximum power from 

 it, when the load resistance is equal to the generator internal resistance. 

 This is the maximum power transfer theorem. It does not mean that a 

 generator delivers maximum power to a given load when the generator 

 internal resistance is made equal to the load. 



Easily solved resistance networks 



The rules for resistances in parallel and in series can be used to solve 

 completely a certain class of resistance network, which may be quite elaborate. 



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