Complex Arithmetic

Introduction

Each problem in this problem set presents an equation involving complex numbers and variables.

Use complex arithmetic to determine the values of the variables.

Appendix B of Introduction to Electric Circuits by R.C. Dorf and J.A Svoboda provides a review

of complex arithmetic.

Worked Examples

Example 1

Given

4565

15 8

j je Aej

Find the values of A and .

Solution:

45 15245 45 107

152

6 6 305 5 1.76

15 8 17 17

jj j

je e e

j e

je

Example 2:

Given:

45102.36 je

a j b

Find the values of a and b.

Solution:

45

45

104.24 3 3

2.36

j

je j a

ej b

Example 3

Given:

3 8 3j 2Ae j j

Find the values of A and .

Solution:90

90 111 21

111

32 32 323.75

3 8 8.54 8.54

jjj j

j

j eAe e e

j e

Example 4:

Given:1352 3 5j jAe j e

Find the values of A and .

Solution:135

161

2 3 5 2 3 3.54 3.54

2 3.54 3 3.54

1.54 0.54

1.63

j j

j

Ae j e j j

j j

j

e

Example 5:

Given:

15

4 3

2

j

j

jAe

e

Find the values of A and .

Solution:37

37 15 52

15 15

4 3 5 52.5

2 2 2

jjj j

j j

j eAe e e

e e

Example 6:

Given:

135

5 13

6 j

ja j b

e

Find the values of a and b.

Solution:111

111 135

135 135

24

5 13 13.9 13.9

6 6 6

2.32 2.12 0.94

jj

j j

j

j ea j b e

e e

e j

Example 7:

Given:

64 3

jj

a j b

Find the values of a and b.

Solution:90

90 143

143

53

6 6 6

4 3 5 5

1.2 0.722 0.958

jj

j

j

j ea j b e

j e

e j

Example 8

Given:120 156 4 3 2j je j e a j b

Find the values of a and b.

Solution:

120 120

120 120

120 120 240

6 4 3 1.93 0.52 6 2.07 3.52

6 4.08

6 4.08 24.48

12.2 21.2

j j

j j

j j

a j b e j j e j

e e

e e

j

Example 9:

Given:120 4 3jAe j b j

Find the values of A and b.

Solution:120 4 3

0.5 0.866 4 3

jAe j b j

A j A j b j

Equating real and imaginary parts:

0.5 4 8A A

and

0.866 3 3 0.866 8 3.93A b b

Example 10:

Given:1206 4 8j je j b e 18

Find the values of A and .

Solution:

120

120

120

6 4 8 18

184 8cos 8sin 3 1.5

6

j j

j

j

e j b e

j b j e je

2.6

Equating real and imaginary parts:

8 cos 2.5 71.8

and

8 sin 2.6 2.6 8 cos(71.8 ) 10.2b b

Example 11:

Given:604 2 2 ja j j Ae

Find the values of a and A.

Solution:604 2 2

8 2 2 0.5 0.866

10 2 0.5 0.866

ja j j Ae

j a A j A

j a A j A

Equating real and imaginary parts:

10 0.5 20A A

and

2 0.866 20 8.66a a