THE USE OF COMPLEX QUANTITY. 91 



Admittance, conductance, and susceptance. By comparing equations (vii) and 

 (viii) it is evident that impedance R +-JX used as an inverse operator is equivalent 

 to the expression Rf (R* + X 2 ) jX/(R 2 -f X 2 } used as a direct operator. This 

 operator is called the admittance of the circuit ; the factor RJ (R 2 -|- X*} which, mul- 

 tiplied by E, gives the component of / parallel to , is called the conductance of 

 the circuit ; and the factor Xj (R 2 -{- X*} which, multiplied by E gives the com- 

 ponent of / 90 behind E in phase, is called the susceptance of the circuit. There- 

 fore, using g for conductance and b for susceptance we have 



r> 



b = 



and, of course 



f=(g-jb}E (xi) 



It is usual to represent the impedance of a circuit as a complex quantity by the 

 letter Z and to represent the numerical value of impedance by the letter z ; and it 

 is also usual to represent the admittance of a circuit as a complex quantity by the 

 letter Y, and to represent the numerical value of admittance by the letter y. Thus 



(xi) 



(xiii) 

 (xiv) 



Further practical information concerning the use of complex quantity in alterna- 

 ting-current problems is given in Arts. 44-47 



43. DeMoivre's Theorem. The use of complex quantity in trigonometric 

 transformations and the correctness of the formulas given in the foregoing article 

 may be most elegantly shown by means of the relation : 



(i) 



which was first pointed out by DeMoivre, e being the Napierian base. This 

 relation may be established by writing j0 for x in the infinite series for e* 

 and separating the terms which do not contain j as a factor (= series for 

 cos 6) from the terms which do contain / as a factor (= series for sin 0). 



(a) Regarding trigonometric transformations. Examples. Squaring both 

 members of equation (i) we have : 



j ' 2e = cos 2 sin* + 2j( sin cos 0) ( ii) 



but, according to equation (i) we must have : 



e^ 2d z=cos20+/sin20 (Hi) 



Therefore, comparing equations (ii) and (Hi), we have : 



cos 20 = cos 2 sin 2 

 and sin 20 = 2 sin cos 



