RESISTANCES 



and / sin cot. This will not be true of circuits in the next chapter, where the 

 instantaneous states of circuits will be found to depend on their past histories, 

 and we shall have to investigate the behaviour of the networks both when 

 connected to direct and alternating supplies. 



Before ending this chapter we have to touch on what is meant by an R.M.S. 

 value in connection with alternating currents. What does it mean to buy an 

 electric lamp for, say, a 230 V a.c. supply, when in fact the voltage is changing 

 the whole time ? Simply this : that if the lamp were connected to a d.c. 

 supply at 230 V, the amount of Hght that would be given off is the same (that 

 is, the filament temperature would be the same) as when the lamp is connected 

 to the 230 V a.c. supply. In other words, a 230 Y a.c. supply is one which 

 produces the same heating effect as a 230 V d.c. supply. 



In Figure 2.41 the instantaneous rate of heat production is equal to the 



ryj") yv= V sin cot 



RMS. voltage-7 



Figure 2.41 



power into the lamp, which is v^jR. Therefore the average rate of heat 

 production is (mean of v^jR). If this average rate could be produced by a 

 direct voltage numerically equal to V, then the rate of heat production 

 would be V^IR and V^jR = (mean of v^jR) 



so K = (mean of v^)^'^ 



— the Root of the Mean Square of v 



For a pure sine wave generator output, v = Ksin a>t, the R.M.S. value V 

 works out to be 0-707 V. Note that this is not the same as the average of 

 V sin cot, which is 0, or even the average of V sin cot over half a cycle, which 

 is (2/7r)K== 0-637 V. 



24 



