THE USE OF COMPLEX QUANTITY. 97 



Determination of the secondary current / 2 . The given primary current induces 

 in the secondary coil an electromotive force which is at each instant equal to 

 - M> dijdt. 



This electromotive force is 90 behind I v its effective value is w MI^ , and its 

 symbolic expression is juMI^ and this electromotive force produces in the second- 

 ary coil a current 



j a m l 



~ 



Reaction of the secondary current upon the primary coil. The secondary cur- 

 rent 7 2 induces in the primary coil an electromotive force which is at each instant 

 equal to M- di^dt. This electromotive force is 90 behind 7 2 in phase, its 

 effective value is uMI v and its complex expression is juMI T The electromotive 

 force induced in the primary by the secondary current must be overcome by the elec- 

 tromotive force which acts upon the primary. The portion of the acting electro- 

 motive force which thus balances the reaction of the secondary current is equal to this 

 reaction and opposite to it in sign and is, therefore, equal to -\-juMI^. 



Determination of total electromotive force acting on primary. This total electro- 

 motive force consists of three parts as follows : 



1. The part described above which balances the reaction of the secondary current. 

 This part is equal to -f- juMI 2 , or using the value of 7 2 from equation (z), we have 

 for this part of the total electromotive force : 



+ 



2. The part used to overcome the resistance of the primary coil. This is at each 

 instant equal to r^, its effective value is r-J^ and its complex expression is r^ 

 since it is in phase with I r 



3. The part used to overcome the inductance of the primary coil. This is at each 

 instant equal to L^'di^dty it is 9Oahead of 7 1} its effective value is wZ^, and 

 its complex expression is juL^. Therefore the total electromotive force required to 

 maintain the given primary current is : 



or, separating components ; 



This equation shows that the effect of the secondary coil is to make the primary coil 

 behave as if its resistance were : 



and its inductance were : 



and these expressions are called the equivalent resistance and the equivalent inductance, 

 respectively, of the system represented in Fig. 89. Compare Art. 28. 

 8 



