Solved Problems

1. Conversion Between Forms
37.
a) 4 + 𝑗3
𝑎 = 4, 𝑏 = 3
𝑟 = √42 + 322
= 5.00
𝜃 = tan−1
3
4
= 36.87°
𝑠𝑜4 + 𝑗3 = 5.00∠36.87°
b) 2 + 𝑗2
𝑎 = 2, 𝑏 = 2
𝑟 = √22 + 222
= 2.83
𝜃 = tan−1
2
2
= 45.00°
𝑠𝑜 2 + 𝑗2 = 2.83∠45.00°
c. 4 + 𝑗12
𝑎 = 4, 𝑏 = 12
𝑟 = √42 + 1222
= 12.65
𝜃 = tan−1
12
4
= 71.57°
𝑠𝑜 4 + 𝑗12 = 12.65∠71.57°
38.
a) −8 − 𝑗16
𝑎 = −8, 𝑏 = −16
𝑟 = √(−8)2 + (−16)22
= 17.89
𝜃 = tan−1
−16
−8
= 243.43°
𝑠𝑜 − 8 − 𝑗16 = 17.89∠243.43°
b) +8 − 𝑗4
𝑎 = +8, 𝑏 = −4
𝑟 = √(8)2 + (−4)22
= 8.94
𝜃 = tan−1
−4
8
= 26.57°
𝑆𝑖𝑛𝑐𝑒 + 8 − 𝑗4 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡, 𝜃 = 360 − 26.57 = 333.43°

2. 𝑠𝑜 + 8 − 𝑗4 = 8.94∠333.43°
c) 0.02−𝑗0.003
𝑎 = 0.02, 𝑏 = −0.003
𝑟 = √(0.02)2 + (−0.003)22
= 0.0202
𝜃 = tan−1
−0.003
0.02
= 8.53°
𝑠𝑜 0.02 − 𝑗0.003 = 0.0202∠8.53°
39. Convert from polar to rectangular
a) 6∠40°
6.00(cos40°
+ 𝑗 sin 40°
)
= (6.00 cos40°
)+ (𝑗6.00sin 40°
)
= 4.60 + 𝑗3.86
b) 12∠120°
12.00(cos120°
+ 𝑗 sin 120°
)
= (12.00cos120°
)+ (𝑗12.00sin 120°
)
= −6.00 + 𝑗10.39
c) 2000∠−90°
2000(cos270°
+ 𝑗 sin 270°
)
= (2000cos270°
)+ (𝑗2000sin 270°
)
= 0.00 − 𝑗2000
40. Convert from polar to rectangular
a) 42∠0.15°
42.00(cos0.15°
+ 𝑗 sin 0.15°
)
= (42.00cos0.15°
)+ (𝑗42.00sin 0.15°
)
= 42 + 𝑗0.11
b) 2002∠−60°
2002(cos300°
+ 𝑗 sin 300°
)
= (2002cos300°
)+ (𝑗2002sin 300°
)
= 1001 + −𝑗1733.78
c) 0.006∠−120°
0.006(cos240°
+ 𝑗 sin 240°
)
= (0.006cos240°
)+ (𝑗0.006sin 240°
)
= −0.003 + −𝑗5.2 × 10−3
Section 14.9

3. 41.
a) (4.2 + 𝑗6.8) + (7.6 + 𝑗0.2)
𝑌𝑜𝑢 𝑎𝑑𝑑 𝑡ℎ𝑒 𝑟𝑒𝑎𝑙 𝑝𝑎𝑟𝑡𝑠 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒𝑙𝑦 𝑡ℎ𝑒𝑛 𝑎𝑑𝑑 𝑡ℎ𝑒 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑝𝑎𝑟𝑡𝑠
= (4.2 + 7.6) + 𝑗(6.8 + 0.2)
= 11.8 + 𝑗7.00
b) (142 + 𝑗7)+ (9.8 + 𝑗42)+ (0.1 + 𝑗0.9)
= (142 + 9.8 + 0.1) + 𝑗(7 + 42 + 0.9)
= 151.9 + 𝑗49.9
c) (4 × 10−6
+ 𝑗76)+ (7.2 × 10−7
− 𝑗5)
= (4 × 10−6
+ 7.2 × 10−7) + 𝑗(76 − 5)
= 4.72 × 10−6
+ 𝑗71
42.
a) (9.8 + 𝑗6.2) − (4.6 + 𝑗4.6)
= (9.8 − 4.6) + 𝑗(6.2 − 4.6)
= 5.2 + 𝑗1.6
b) (167 + 𝑗243)− (−42.3 − 𝑗68)
= (167 + 42.3) + 𝑗(243 + 68)
= 209.3 + 𝑗311
c) (−36 + 𝑗78)− (−4 − 𝑗6) + (10.8 − 𝑗72)
= (−36 + 4 + 10.8) + 𝑗(78 + 6 − 72)
= −21.2 + 𝑗12
43.
a) 6∠20°
+ 8∠80°
= 12.16∠54.72°
b) 42∠45°
+ 62∠60°
− 70∠120°
= 146.3∠79.88°
c) 20∠−120°
− 10∠−150°
+ 8∠−210°
+ 8∠+240°
= 38.89∠139.53°
44. Perform the following multiplication in rectangular form
a) (2 + 𝑗3)(6+ 𝑗8)
= (2 × 6 − 3 × 8) + 𝑗(6 × 3 + 2 × 8)
= −12 + 𝑗34
b) (7.8 + 𝑗1)(4+ 𝑗2)(7+ 𝑗6)

4. = 86.8 + 𝑗312.4
c) (400 − 𝑗200)(−0.01 − 𝑗0.5)(−1 + 𝑗3)
= 698 + 𝑗114
45. Perform the following multiplications in polar form
a) (2∠60°
)(4∠ − 40°
)
= (2 × 4)∠(60− 40)
8∠20°
b) (6.9∠8°
)(7.2∠ − 72°
= (6.9 × 7.2)∠(8− 72)
849.68∠−62°
c) (0.002∠120°
)(0.5∠200°
)(40∠80°
= (0.002 × 0.5 × 40)∠(120+ 200 + 80)
0.04∠400°
46. Perform the following divisions in polar
a)
42∠10°
7∠ 60°
=
42
7
∠10 − 60
6∠−50°
b)
0.006∠120°
30∠ 60°
=
0.006
30
∠120 − 60
2 × 10−4
∠60°
c)
4360∠−20°
40∠−210°
=
4360
40
∠ − 20 + 210
109∠190°
47. Perform the following divisions and leave the answer in rectangular form
a)
(8+𝑗8)
(2+𝑗2)
(8 + 𝑗8)
(2 + 𝑗2)
= 4

5. Section 14.11 Phasors
52. Express the following in phasor form
a) √2(160)sin( 𝜔𝑡 + 30°)
= 226.27∠30°
b) √2(25 × 10−3)sin(157𝑡 − 40°)
= 0.0354∠−40°
c) 100 sin( 𝜔𝑡 − 90°)
= 100∠270°
53. Express the following in phasor form
a) 20 sin(377𝑡 − 180°)
= 20∠−180°
b) 6 × 10−6
cos 𝜔𝑡
6 × 10−6
c) 3.6 × 10−6
cos(745𝑡 − 20°)
= 3.6 × 10−6
∠−20°
54. Express the following phasor currents and voltages as sine waves if the frequency is 60 Hz.
a) 𝐼 = 40𝐴∠20°
𝐼 = 40cos(2𝜋 × 60𝑡 + 20°)
𝐼 = 40cos(120𝜋𝑡 + 20°) 𝐴
b) 𝑉 = 120𝑉∠10°
𝑉 = 120cos(2𝜋 × 60𝑡 + 10°)
𝐼 = 120cos(120𝜋𝑡 + 10°) 𝑉
c) 𝐼 = 8 × 10−3
𝐴∠−110°
𝐼 = 40sin(2𝜋 × 60𝑡 − 110°)
𝐼 = 40sin(120𝜋𝑡 − 110°) 𝐴
d) 𝑉 =
6000
√2
𝑉∠ − 180°
𝑉 = 4242.64sin(2𝜋 × 60𝑡 − 180°)
𝐼 = 4242.64cos(120𝜋𝑡 − 180°) 𝑉