v^ • cos9^ + Vg • cosSg = Vq • cos9^ (C-5) 



v^ • s1q9^ ■•■ ^B * 3ln6g = Vq • slnS^ (C-6) 



Squaring both sides and summing equations C-5 and C-6 removes 

 the e^, teirras (cos^e + sln^e = 1), 



^C^ " ^A^ • cos^e^ + V32 . cos^Sg + 2 • v^ • Vg • cos9^ • cosBg + 

 v^^ • aln^e^ + Vg2 • 3ln^9g + 2 • v^ • Vg • slnS^ • slnSg (c-7) 



2 2 



Again using the relationship cos 9 + sin 9=1, the addition 



formula for cosines, and taking the root of the result gives the 



formula for the Vq term as, 



''C =• ^V + V + 2 • V;^ • vg • (cos(9^-9g)//2 (c-8) 



The unknown resulting phase angle Bq can be derived by dividing 

 (C-6) by (C-5), 



^A * 3^°®A "^ ''B * 3^"S 

 tan9g = (C-9) 



^A * ^^'^^A "^ ^B * "^^B 



_ v^ • sme^ + vg . sln9 

 9c = tan ^ [ ) (C-IO) 



^A * "89^ + Vg • cos9g 



179 



