THE MEASUREMENT OF CURRENT 79 



to the mean product of the currents in the two coils. Therefore, 

 if the coils are in series the deflection is proportional to the mean 

 square of the current or to the square of the effective value. 

 The currents in the two coils may differ in wave form and in time 

 phase as well as in magnitude. For instance, expressing both i p 

 and i M in the form of a series, if the waves are non-sinusoidal, 



IP = A\ sin ut + A% sin (2oo2 <p 2 ) + ^.3 sin 

 iu = BI sin (ut <f>'\) + Bz sin (2ut <p' 2 ) 



B 3 sin 



It is well known that the mean product of two sine curves 

 which differ in periodicity is zero, and that the mean product of 



two sine curves of the same periodicity is -= cos a where 7i and 



/2 are the maximum values and a is the angle of phase difference. 

 Consequently, 



rO = KI^ \ ipi M dt JCij g cos ^'i + 9 oos (^2 <f>'z) + 



cos (v> 3 - *',) + . . . (63) 



If the fixed and movable coils are in series, as they are when 

 currents are measured, A n = B n = 7 n and cos (<? tf>' n ) = 1, so 



KI [T + T + T+ -] =Ki12 (64) 



re 



/i, /2, /s, etc., are the maximum values of the various compo- 



/i 2 

 nents. -^ is the square of the effective value of the fundamental, 



-g- is the square of the effective value of the second harmonic, 



etc., and 7 2 is the square of the effective value of the current. 



2. When the movable coil is allowed to deflect, the factor KI be- 

 comes a variable depending on the angle between the axes of the 

 fixed and movable coils. If the field due to the fixed coil is uni- 

 form or, what practically amounts to the same thing, if the mov- 

 able coil is very much smaller than the fixed coil, K\ is propor- 

 tional to the cosine of the angle of displacement of the axes of the 



