APPENDIX
F
Complex Numbers
j=√
− 1 = 1 90 ◦;j^2 =− 1 = 1 180 ◦;j^3 =−j= 1 270 ◦= 1 − 90 ◦;j^4 = 1 0 ◦e±jθ=cosθ±jsinθ= 1 ±θejθ+e−jθ=2 cosθejθ−e−jθ=j2 sinθIfA ̄=Aθ=Aejθ=a+jb,then
a=Acosθ=ReA ̄;b=Asinθ=ImA ̄
A=√
a^2 +b^2 ;θ=tan−^1 (b/a)
(A) ̄∗=A −θ=Ae−jθ=a−jb
A ̄+(A) ̄∗=a+jb+a−jb= 2 a=2ReA ̄
A( ̄A) ̄∗=Aejθ×Ae−jθ=A^2
IfA ̄ 1 =a 1 +jb 1 and A ̄ 2 =a 2 +jb 2 ,it follows that
A ̄ 1 ±A ̄ 2 =(a 1 +jb 1 )±(a 2 +jb 2 )=(a 1 ±a 2 )+j(b 1 ±b 2 )
Re{A ̄ 1 ±A ̄ 2 }=ReA ̄ 1 ±ReA ̄ 2
Im{A ̄ 1 ±A ̄ 2 }=ImA ̄ 1 ±ImA ̄ 2
A ̄ 1 A ̄ 2 =(a 1 +jb 1 )(a 2 +jb 2 )=(a 1 a 2 −b 1 b 2 )+j(a 1 b 2 +a 2 b 1 )
A ̄ 1
A ̄ 2=a 1 +jb 1
a 2 +jb 2=a 1 +jb 1
a 2 +jb 2×a 2 −jb 2
a 2 −jb 2=(a 1 a 2 +b 1 b 2 )+j(a 2 b 1 −a 1 b 2 )
a^22 +b 22
IfA ̄ 1 =A 1 ejθ^1 =A 1 θ 1 and A ̄ 2 =A 2 ejθ^2 =A 2 θ 2 ,then
A ̄ 1 A ̄ 2 =(A 1 ejθ^1 )(A 2 ejθ^2 )=(A 1 θ 1 )(A 2 θ 2 )=A 1 A 2 ej(θ^1 +θ^2 )=A 1 A 2 θ 1 +θ 2
A ̄ 1
A ̄ 2=
A 1 ejθ^1
A 2 ejθ^2=
A 1 θ 1
A 2 θ 2=
A 1
A 2ej(θ^1 −θ^2 )=
A 1
A 2 θ 1 −θ 2ASSOCIATIVE, COMMUTATIVE, AND DISTRIBUTIVE LAWS ARE SUMMARIZED
A ̄+(B ̄+C) ̄ =(A ̄+B) ̄ +C ̄;(A ̄B) ̄ C ̄=A( ̄B ̄C) ̄
A ̄+B ̄=B ̄+A ̄;A ̄B ̄=B ̄A ̄
A( ̄B ̄+C) ̄ =A ̄B ̄+A ̄C ̄846