10 ALTERNATING CURRENTS 



value of this squared wave is 1.0 amp., as shown by the dotted 

 1 Because the areas above the dotted line will just fit into 

 the shaded valleys below the dotted line. Therefore, if an 

 equivalent rectangle were made from this wave, its height 

 would be 1.0 unit. This value, 1.0, is the average of the squares 

 of the current wave. Average heating varies as the average of 

 the squares of the current, so this procedure for determining the 

 ampere value of the wave of Fig. 7 (a) is correct. 



To obtain the correct value of the current in amperes, the 

 square root of the average square must be taken. That is, I (in 

 amperes) = \/1.0 = 1.0 amp. This value of the current is called 

 the root-mean-square (r.m.s.) or effective value of the current. 



Therefore, an alternating-current ampere, sine wave, which 

 produces heat at the same rate as a direct-current ampere, has 

 a maximum value of 1.414 (= \/2) amp. In fact, for any sine- 

 wave current, the ratio of the maximum to the effective value is 

 equal to the \/2 or 1.414. The ratio of effective to maximum 

 value is 1/1.414 = 0.707. 



To obtain the effective value of any current wave, not necessarily 

 a sine wave: 



(a) Plot a wave whose ordinates are equal to the squares of the 

 ordinates of the given current wave. 



(6) Find the average value of this squared wave by obtaining 

 the area of its loops with a planimeter and dividing this area by 

 the base; or by averaging the ordinates. 



(c) Find the square root of this average. 



The same result may be obtained by erecting equidistant 

 ordinates on the original wave, averaging their squares and 

 taking the square root of this average. This will give the root- 

 mean-square value. 



If a sine wave of current be averaged in the ordinary manner for 

 half a cycle, it will be found that this average is equal to 2/w or 

 0.637 times the maximum value. The ratio of effective to 

 average value is then 0.707/0.637 = 1.11 and the ratio of 

 average to effective value is 0.9. It is sometimes necessary 

 to know the average value, and the ratio of effective to average 

 value enters into computations of induced emfs. in alternators, 

 transformers and other types of alternating-current machinery. 



The ratio of effective to average value is called the form factor 



