A
Exercises APPENDIX EA.1 Given Z 1 = 2 j 3 and Z 2 = 8 + j 6, we have:
Z 1 + Z 2 = 10 + j 3 Z 1 Z 2 = 6 j 9
Z 1 Z 2 = 16 j 24 + j 12 j 2 18 = 34 j 12
2 j 3 8 j 6 16 j 12 j 24 + j 2 18 Z1 / Z2 = = = 0.02 j 0.36 8+ j6 8 j6 100
EA.2
Z 1 = 1545 o = 15 cos( 45 o ) + j 15 sin( 45 o ) = 10.6 + j 10.6 Z 2 = 10 150 o = 10 cos( 150 o ) + j 10 sin( 150 o ) = 8.66 j 5 Z 3 = 590 o = 5 cos(90 o ) + j 5 sin(90 o ) = j 5
Notice that Z1 lies in the first quadrant of the complex plane.
EA.3
Z 1 = 3 + j 4 = 32 + 4 2 arctan( 4 / 3) = 553.13o
Notice that Z2 lies on the negative imaginary axis. Z 2 = j 10 = 10 90 o Notice that Z3 lies in the third quadrant of the complex plane.
Z 3 = 5 j 5 = 52 + 52 (180 o + arctan( 5 / 5)) = 7.07 225 o = 7.07 135 o
EA.4
Notice that Z1 lies in the first quadrant of the complex plane.
Z 1 = 10 + j 10 = 10 2 + 10 2 arctan(10 / 10) = 14.1445 o = 14.14 exp( j 45 o )
Notice that Z2 lies in the second quadrant of the complex plane.
Z 2 = 10 + j 10 = 10 2 + 10 2 (180 o + arctan( 10 / 10))
= 14.14135 o = 14.14 exp( j 135 o )
1
EA.5
Z 1Z 2 = (1030 o )(20135 o ) = (10 20)(30 o + 135 o ) = 200(165 o ) Z 1 / Z 2 = (1030 o ) /(20135 o ) = (10 / 20)(30 o 135 o ) = 0.5( 105 o )
Z 1 Z 2 = (1030 o ) (20135 o ) = (8.66 + j 5) ( 14.14 + j 14.14)
= 22.8 j 9.14 = 24.6 21.8o
Z 1 + Z 2 = (1030 o ) + (20135 o ) = (8.66 + j 5) + ( 14.14 + j 14.14)
= 5.48 + j 19.14 = 19.9106o
Problems PA.1
Given 1 Z = 2 + j 3 and Z 2 = 4 j 3, we have:
Z1 + Z2 = 6 + j 0 Z 1 Z 2 = 2 + j 6 Z 1 Z 2 = 8 j 6 + j 12 j 2 9 = 17 + j 6
Z1 / Z2 =
2 + j 3 4 + j 3 1 + j 18 = = 0.04 + j 0.72 4 j3 4 + j3 25
PA.2
Given that Z 1 = 1 j 2 and Z 2 = 2 + j 3, we have:
Z1 + Z2 = 3 + j 1 Z 1 Z 2 = 1 j 5 Z1 Z2 = 2 + j 3 j 4 j 2 6 = 8 j 1
Z1 / Z2 =
1 j2 2 j3 4 j 7 = = 0.3077 j 0.5385 2 + j3 2 j3 13
2
PA.3
Given that Z 1 = 10 + j 5 and Z 2 = 20 j 20, we have:
Z 1 + Z 2 = 30 j 15 Z 1 Z 2 = 10 + j 25 Z 1 Z 2 = 200 j 200 + j 100 j 2 100 = 300 j 100
Z1 / Z2 =
10 + j 5 20 + j 20 100 + j 300 = = 0.125 + j 0.375 20 j 20 20 + j 20 800
PA.4
(a) (b) (c) (d)
Z a = 5 j 5 = 7.071 45o = 7.071 exp( j 45o ) Z b = 10 + j 5 = 11.18153.43o = 11.18 exp(j 153.43o ) Z c = 3 j 4 = 5 126.87 o = 5 exp( j 126.87 o ) Z d = j 12 = 12 90 o = 12 exp( j 90 o ) Z a = 545o = 5 exp(j 45o ) = 3.536 + j 3.536 Z b = 10120 o = 10 exp(j 120 o ) = 5 + j 8.660 Z c = 15 90 o = 15 exp( j 90 o ) = j 15 Z d = 1060 o = 10 exp( j 120 o ) = 5 j 8.660
PA.5
(a) (b) (c) (d)
PA.6
(a) (b) (c)
Z a = 5e j 30 = 530 o = 4.330 + j 2.5
o
Z b = 10e j 45 = 10 45o = 7.071 j 7.071
o
Z c = 100e j 135 = 100135o = 70.71 + j 70.71
o
3
(d)
Z d = 6e j 90 = 690 o = j 6
o
PA.7
(a) (b) (c)
Z a = 5 + j 5 + 1030 o = 13.66 + j 10 Z b = 545o j 10 = 3.536 j 6.464
Zc =
1045 o 1045 o 2 8.13o = 1.980 j 0.283 = 3 + j4 553.13o
(d)
Zd =
15 = 3 90 o = j 3 o 590
4