Upload
jota-eduardo-rojas
View
5
Download
0
Embed Size (px)
Citation preview
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