136 ELECTRICAL ENGINEERING 



To distinguish between horizontal and vertical components 

 the prefix j = Vl is added to all vertical components and 

 the expression for the e.m.f. above is 



E = e l +je z ........ (190) 



The dot is placed under the E to show that it is expressed in rec- 

 tangular coordinates and serves to distinguish it from its absolute 

 value. 



Since e\ = E cos < = Ir and e^ = E sin < = Ix, 



E = E cos <f> + jE sin 



= I(r+jx), .......... (191) 



and therefore the impedance in rectangular coordinates is 



z = r+jx ........ (192) 



In Fig. 103 the e.m.f. is chosen as axis and the current is behind 

 it in phase by an angle </> and has two components i\ in phase with 

 the e.m.f. and iz in quadrature behind it. 



The current may be written 



/ =ii-jit, ........ (193) 



and this equation indicates that the current has a value 



and that it is behind the chosen axis (in this case the e.m.f.) in 

 phase by an angle $, where 



tan< = ? 



Since i\ = 7 cos < = Eg and iz = I sin $ = Eb, where y = 

 Vgf 2 + 6 2 is the admittance of the circuit, equation 193 may 

 be written 



1=1 cos <f> jl sin 

 = Eg - jEb 

 = E(g-jb), ...... . (194) 



and therefore the admittance in rectangular coordinates is 



y = g- j&. 



Admittance and impedance are not alternating quantities and 

 their components are independent of the axis of reference but 

 they can be represented in rectangular coordinates as shown. 



